<PAGE>
 
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                        BELLWETHER EXPLORATION COMPANY
                            a Delaware Corporation
                                (the "Company")

                                    WARRANT

                              To Purchase 100,000

                     Shares of the Company's Common Stock

                                   issued to

                      Torch Energy Advisors Incorporated,
 
                             (the "Warrantholder")


                                 April 9, 1997


          This Warrant and any Shares acquired upon the exercise of this 
     Warrant have not been registered under the Securities Act of 1933, 
     as amended, and may not be transferred, sold or otherwise disposed 
     of in the absence of such registration or an exemption therefrom 
     under such Act. This Warrant and such Shares may be transferred 
     only in compliance with the conditions specified in this Warrant.

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                        BELLWETHER EXPLORATION COMPANY

                                    Warrant

No. W-001                                                          April 9, 1997


     BELLWETHER EXPLORATION COMPANY, a Delaware corporation (the "Company"), for
value received, hereby certifies that Torch Energy Advisors Incorporated, a
Delaware corporation (the "Purchaser"), or its registered assign(s) (along with
Purchaser, each a "holder"), is entitled to purchase from the Company 100,000
duly authorized, validly issued, fully paid and nonassessable shares of Common
Stock, $.01 par value (the "Common Stock"), at any time or from time to time
prior to 5:00 p.m., Houston, Texas, time, on the Expiration Date, all subject to
terms, conditions and adjustments set forth in this warrant ("Warrant").

     Certain capitalized terms used in this Warrant are defined in Article VII;
unless otherwise specified, references to an "Exhibit" mean one of the exhibits
attached to this Warrant, references to an "Article" mean one of the articles in
this Warrant and references to a "Section" mean one of the sections of this
Warrant. This Warrant is issued pursuant to the letter agreement dated February
5, 1997, between Purchaser and the Company entitling the holder to receive
warrants to purchase an aggregate of 100,000 shares of Common Stock.

                        ARTICLE I.  EXERCISE OF WARRANT

     Section 1.1.  Manner of Exercise. (a) Subject to Subsection (b) of this
Section 1.1, this Warrant may be exercised by the holder hereof, in whole or in
part, during normal business hours on any Business Day, by surrender of this
Warrant to the Company at its office maintained pursuant to subdivision (a) of
Section 6.2, accompanied by a subscription in substantially the form attached to
this Warrant (or a reasonable facsimile thereof) duly executed by such holder
and accompanied by payment, in cash or by certified or official bank check
payable to the order of the Company in the amount obtained by multiplying (i)
the number of shares of Common Stock (without giving effect to any adjustment
thereof) designated in such subscription by (ii) the Warrant Price, and such
holder shall thereupon be entitled to receive the number of duly authorized,
validly issued, fully paid and nonassessable shares of Common Stock (or Other
Securities) determined as provided in Articles II through IV.

     (b)  In lieu of delivering the number of shares of Common Stock calculated
under subsection (a) of this Section 1.1, The
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Company may, at its election, and shall if requested by the holder of this
Warrant, issue to the holder of this Warrant upon exercise a number of duly
authorized, validly issued, fully paid and non-assessable shares of Common Stock
equal to the following, rounded to the nearest whole share: the quotient of (i)
the product of (x) the number of shares of Common Stock to be delivered under
such subsection (a) multiplied by the Market Price of the Common Stock on the
date of exercise minus (y) the amount the holder is required to pay to the
Company under such subsection (a) upon such exercise, divided by (ii) the Market
Price of the Common Stock on the date of exercise. If the Company delivers
shares of Common Stock under this subsection (b), then the holder shall not be
required to make any payment in connection with the exercise of this Warrant.

     Section 1.2.  When Exercise Effective. Each exercise of this Warrant shall
be deemed to have been effected immediately prior to the close of business on
the Business Day on which this Warrant shall have been surrendered to the
Company as provided in Section 1.1, and at such time the Person or Persons in
whose name or names any certificate or certificates for shares of Common Stock
(or Other Securities) shall be issuable upon such exercise as provided in
Section 1.3 shall be deemed to have become the holder or holders of record
thereof.

     Section 1.3.  Delivery of Stock Certificates, etc. As soon as practicable
after each exercise of this Warrant, in whole or in part, and in any event
within five Business Days thereafter, the Company, at its expense (including the
payment by it of any applicable issue taxes), will cause to be issued in the
name of and delivered to the exercising holder hereof, subject to Article V, as
such exercising holder (upon payment by such exercising holder of any applicable
transfer taxes) may direct, the following:

          (a)  Certificates. A certificate or certificates for the number of
     duly authorized, validly issued, fully paid and nonassessable shares of
     Common Stock (or Other Securities) to which such exercising holder shall be
     entitled upon such exercise plus, in lieu of any fractional share to which
     such exercising holder would otherwise be entitled, cash in an amount equal
     to the same fraction of the Closing Price per share on the Business Day
     next preceding the date of such exercise.


                                      -2-
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          (b)  Warrant. In case such exercise is in part only, a new Warrant or
     Warrants of like tenor dated the date hereof, calling in the aggregate on
     the face or faces thereof for the number of shares of Common Stock equal
     (without giving effect to any adjustment thereof) to the number of such
     shares called for on the face of this Warrant minus the number of such
     shares designated by the holder upon such exercise as provided in 
     Section 1.1.

         ARTICLE II. ADJUSTMENT OF COMMON STOCK ISSUABLE UPON EXERCISE

     Section 2.1.  General; Warrant Price. The number of shares of Common Stock
which the holder of this Warrant shall be entitled to receive upon each exercise
hereof shall be determined by multiplying the number of shares of Common Stock
which would otherwise (but for the provisions of this Article II) be issuable
upon such exercise, as designated by the holder hereof pursuant to Section 1.1,
by a fraction (a) the numerator of which is the Initial Price and (b) the
denominator of which is the Warrant Price in effect at the effective time of
such exercise (as provided in Section 1.2). The "Warrant Price" shall initially
be the Initial Price, shall be adjusted and readjusted from time to time as
provided in this Article II and, as so adjusted or readjusted, shall remain in
effect until a further adjustment or readjustment thereof is required by this
Article II.

     Section 2.2.  Adjustment of Warrant Price.

     (1)  In case the Company shall pay or make a dividend or other distribution
on its Common Stock exclusively in Common Stock (or Other Securities) or shall
pay or make a dividend or other distribution on any other class of capital stock
of the Company which dividend or distribution includes Common Stock (or Other
Securities), the Warrant Price in effect at the opening of business on the day
next following the date fixed for the determination of stockholders entitled to
receive such dividend or other distribution shall be reduced by multiplying such
Warrant Price by a fraction of which the numerator shall be the number of shares
of Common Stock (or Other Securities) outstanding at the close of business on
the date fixed for such determination and the denominator shall be the sum of
such number of shares and the total number of shares constituting such dividend
or other distribution, such reduction to become effective immediately after the
opening of business on the day next following the date fixed for such
determination. For the purposes of this Section 2.2, the number of shares of
Common Stock (or Other Securities) at any time outstanding shall not include
shares held in the treasury of the Company.


                                      -3-
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     (2)  In case the Company shall pay or make a dividend or other distribution
on its Common Stock (or Other Securities) consisting exclusively of, or shall
otherwise issue to all holders of its Common Stock (or Other Securities),
rights, warrants or options entitling the holders thereof to subscribe for or
purchase shares of Common Stock (or Other Securities) at a price per share less
than the Market Price per share of the Common Stock (or Other Securities) on the
date fixed for the determination of stockholders entitled to receive such
rights, warrants or options, the Warrant Price in effect at the opening of
business on the day following the date fixed for such determination shall be
reduced by multiplying such Warrant Price by a fraction of which the numerator
shall be the number of shares of Common Stock (or Other Securities) outstanding
at the close of business on the date fixed for such determination plus the
number of shares of Common Stock (or Other Securities) which the aggregate of
the offering price of the total number of shares of Common Stock (or Other
Securities) so offered for subscription or purchase would purchase at such
Market Price and the denominator shall be the number of shares of Common Stock
(or Other Securities) outstanding at the close of business on the date fixed for
such determination plus the number of shares of Common Stock (or Other
Securities) so offered for subscription or purchase, such reduction to become
effective immediately after the opening of business on the day following the
date fixed for such determination.

     (3)  In case outstanding shares of Common Stock (or Other Securities)
shall be subdivided into a greater number of shares of Common Stock (or Other
Securities), the Warrant Price in effect at the opening of business on the day
following the day upon which such subdivision becomes effective shall each be
proportionately reduced, and, conversely, in case outstanding shares of Common
Stock (or Other Securities) shall be combined into a smaller number of shares of
Common Stock (or Other Securities), the Warrant Price in effect at the opening
of business on the day following the day upon which such combination becomes
effective shall be proportionately increased, such reduction or increase, as the
case may be, to become effective immediately after the opening of business on
the day following the day upon which such subdivision or combination becomes
effective.

     (4)  Subject to the last sentence of this paragraph (4), in case the
Company shall, by dividend or otherwise, distribute to all holders of its Common
Stock (or Other Securities) evidences of its

                                      -4-
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indebtedness, shares of any class of capital stock, securities, cash or property
(excluding any rights, warrants or options referred to in paragraph (2) of this
Section 2.2, any dividend or distribution paid exclusively in cash and any
dividend or distribution referred to in paragraph (1) of this Section 2.2), the
Warrant Price shall be reduced by multiplying the Warrant Price in effect
immediately prior to the effectiveness of the Warrant Price reduction
contemplated by this paragraph (4) by a fraction of which the numerator shall be
the Market Price per share of the Common Stock (or Other Securities) on the date
of such effectiveness less the fair market value (as determined in good faith by
the Board of Directors, whose determination shall be conclusive and shall, in
the case of securities being distributed for which prior thereto there is an
actual or when issued trading market, be no less than the value determined by
reference to the average of the closing prices in such market over the period
specified in the succeeding sentence), on the date of such effectiveness, of the
portion of the evidences of indebtedness, shares of capital stock, securities,
cash and property so distributed applicable to one share of Common Stock (or
Other Securities) and the denominator of which shall be the Market Price per
share of Common Stock (or Other Securities), such reduction to become effective
immediately prior to the opening of business on the day next following the later
of (a) the date fixed for the payment of such distribution and (b) the date 10
days after the notice relating to such distribution is given pursuant to Section
4.3 (such later date of (a) and (b) being referred to as the "Reference Date").
If the Board of Directors determines the fair market value of any distribution
for purposes of this paragraph (4) by reference to the actual or when issued
trading market for any securities comprising such distribution, it must in doing
so consider the prices in such market over the same period used in computing the
Market Price per share. For purposes of this paragraph (4), any dividend or
distribution that includes shares of Common Stock (or Other Securities) or
rights, warrants or options to subscribe for or purchase shares of Common Stock
(or Other Securities) shall be deemed instead to be (a) a dividend or
distribution of the evidences of indebtedness, cash, property, shares of capital
stock or securities other than such shares of Common Stock (or Other Securities)
or such rights, warrants or options (making any Warrant Price reduction required
by this paragraph (4)) immediately followed by (b) a dividend or distribution of
such shares of Common Stock (or Other Securities) or such rights, warrants or
options (making any further Warrant Price reduction required by paragraph (1) or
(2) of this Section 2.2, except (i) the Reference Date of such dividend or
distribution as defined in this paragraph (4) shall be substituted as "the date
fixed for the determination of stockholders entitled to receive

                                      -5-
<PAGE>
 
such dividend or other distribution", "the date fixed for the determination of
stockholders entitled to receive such rights, warrants or options" and "the date
fixed for such determination" within the meaning of paragraphs (1) and (2) of
this Section 2.2 and (ii) any shares of Common Stock (or Other Securities)
included in such dividend or distribution shall not be determined "outstanding
at the close of business on the date fixed for such determination" within the
meaning of paragraph (1) of this Section 2.2).

     (5)  In case the Company shall, by dividend or otherwise, make a
distribution to all holders of its Common Stock (or Other Securities)
exclusively in cash in an aggregate amount that, together with the aggregate
amount of any other distributions to all holders of its Common Stock (or Other
Securities) made exclusively in cash within the 365 days preceding the date of
payment of such distribution and in respect of which no Warrant Price adjustment
pursuant to this paragraph (5) has been made, exceeds 12.5% of the product of
the Market Price per share of the Common Stock (or Other Securities) on the date
fixed for stockholders entitled to receive such distribution times the number of
shares of Common Stock (or Other Securities) outstanding on such date, the
Warrant Price shall be reduced by multiplying the Warrant Price in effect
immediately prior to the effectiveness of the Warrant Price reduction
contemplated by this paragraph (5) by a fraction of which the numerator shall be
the Market Price per share of the Common Stock (or Other Securities) on the date
of such effectiveness less the amount of cash so distributed applicable to one
share of Common Stock (or Other Securities) and the denominator shall be such
Market Price per share of the Common Stock (or Other Securities), such reduction
to become effective immediately prior to the opening of business on the later of
(a) the day following the date fixed for the payment of such distribution and
(b) the date 10 days after the notice relating to such distribution is given.

     (6)  The Company may make such reductions in the Warrant Price, in addition
to those required by paragraphs (1), (2), (3), (4), and (5) of this Section, as
it considers to be advisable in order that any event treated for Federal income
tax purposes as a division of stock or stock rights shall not be taxable to the
recipients.

     (7)  No adjustment in the Warrant Price shall be required unless such
adjustment would require an increase or decrease of at least 1% in the Warrant
Price; provided, however, that any adjustments which by reason of this paragraph
(7) are not required

                                      -6-
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to be made shall be carried forward and taken into account in any subsequent
adjustment.

                   ARTICLE III.  CONSOLIDATION, MERGER, ETC.

     Section 3.1.  Adjustments for Consolidation, Merger, Sale of Assets,
Reorganization, etc. In case the Company after the date hereof (a) shall
consolidate with or merge into any other Person and shall not be the continuing
or surviving corporation of such consolidation or merger, or (b) shall permit
any other Person to consolidate with or merge into the Company and the Company
shall be the continuing or surviving Person but, in connection with such
consolidation or merger, the Common Stock or Other Securities shall be changed
into or exchanged for stock or other securities of any other Person or cash or
any other property, or (c) shall transfer all or substantially all of its
properties or assets to any other Person, (d) shall effect a capital
reorganization or reclassification of the Common Stock or Other Securities
(other than a capital reorganization or reclassifications for which adjustment
in the Warrant Price is provided in Section 2.2), or (e) if permitted by
applicable law, shall effect a statutory share exchange, then, and in the case
of each such transaction, proper provision shall be made so that, upon the basis
and the terms and in the manner provided in this Warrant, the holder of this
Warrant, upon the exercise hereof at any time after the consummation of such
transaction, shall be entitled to receive (at the aggregate Warrant Price in
effect at the time of such consummation for all Common Stock or Other Securities
issuable upon such exercise immediately prior to such consummation), in lieu of
the shares of Common Stock or Other Securities issuable upon such exercise prior
to such consummation, the amount of securities, cash or other property to which
such holder would actually have been entitled as a shareholder upon such
consummation if such holder had exercised the rights represented by this Warrant
immediately prior thereto; provided, however, that if the transaction described
in clauses (a) through (e) hereof provides an election to receive cash, Common
Stock or Other Securities or property, the holder of this Warrant shall, within
10 business days following written request from the Company, notify the Company
of the election such holder would have made had he been a stockholder of the
Company which notice shall govern the consideration to be received upon exercise
of the Warrant, and if no such notice is received within such 10 business days,
the Company in its discretion may determine the consideration to which the
holder of a Warrant is entitled as if the holder had made any of such elections.


                                      -7-
<PAGE>
 
     Section 3.2.  Assumption of Obligations. Notwithstanding anything contained
in this Warrant to the contrary, the Company will not effect any of the
transactions described in clauses (a) through (d) of Section 3.1 unless, prior
to the consummation thereof, each Person (other than the Company) which may be
required to deliver any stock, securities, cash or property upon the exercise of
this Warrant as provided herein shall assume, by written instrument delivered
to, and reasonably satisfactory to, the holder of this Warrant, (a) the
obligations of the Company under this Warrant (and if the Company shall survive
the consummation of such transaction, such assumption shall be in addition to,
and shall not release the Company from, any continuing obligations of the
Company under this Warrant), and (b) the obligation to deliver to such holder
such shares of stock, securities, cash or property as, in accordance with the
foregoing provisions of this Article III, such holder may be entitled to
receive.

               ARTICLE IV.  OTHER PROVISIONS CONCERNING DILUTION

     Section 4.1.  No Impairment. The Company (a) will not permit the par value
of any shares of stock receivable upon the exercise of this Warrant to exceed
the amount payable therefor upon such exercise, (b) will take all such action as
may be necessary or appropriate in order that the Company may validly and
legally issue fully paid and nonassessable shares of Common Stock on the
exercise of the Warrant from time to time outstanding, and (c) will not take any
action which results in any adjustment of the Warrant Price if the total number
of shares of Common Stock (or Other Securities) issuable after the action upon
the exercise of the Warrant would exceed the total number of shares of Common
Stock (or Other Securities) then authorized by the Company's certificate of
incorporation and available for the purpose of issuance upon such exercise.

     Section 4.2.  Accountant's and Company's Report as to Adjustments. In each
case of any adjustment or readjustment in the shares of Common Stock (or Other
Securities) issuable upon the exercise of this Warrant, the Company at its
expense will promptly compute such adjustment or readjustment in accordance with
the terms of this Warrant and cause independent certified public accountants of
recognized national standing (which may be the regular auditors of the Company)
selected by the Company to verify such computation (other than any computation
of the fair value of property as determined in good faith by the Board of
Directors of the Company) and prepare a report setting forth such adjustment or
readjustment and showing in reasonable detail the method of


                                      -8-
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calculation thereof and the facts upon which such adjustment or readjustment is
based, including a statement of (a) the consideration received or to be received
by the Company for any securities issued or sold or deemed to have been issued,
(b) the number of shares of Common Stock outstanding or deemed to be
outstanding, and (c) the Warrant Price in effect immediately prior to such issue
or sale and as adjusted and readjusted (if required by Article II) on account
thereof.  The Company will forthwith mail a copy of each such report to each
holder of a Warrant and will, upon the written request at any time of any holder
of a Warrant, furnish to such holder a like report setting forth the Warrant
Price at the time in effect and showing in reasonable detail the manner in which
it was calculated.

     Section 4.3.  Notices of Corporate Action.  In the event that any of the
following occurs,

          (a)  any taking by the Company of a record of the holders of Common
     Stock (or Other Securities) for the purpose of determining the holders
     thereof who are entitled to receive any dividend or other distribution, or
     any right to subscribe for, purchase or otherwise acquire any shares of
     stock of any class or any other securities or property, or to receive any
     other right, or

          (b)  any capital reorganization of the Company, any reclassification
     or recapitalization of the capital stock of the Company or any
     consolidation, merger or statutory share exchange involving the Company and
     any other Person or any transfer of all or substantially all the assets of
     the Company to any other Person, or

          (c)  any voluntary or involuntary dissolution, liquidation or winding-
     up of the Company,

the Company will mail to each holder of a Warrant a notice specifying (i) the
date or expected date as of which any such record is to be taken for the purpose
of such dividend, distribution or right, and the amount and character of such
dividend, distribution or right, and (ii) the date or expected date on which any
such reorganization, reclassification, recapitalization, consolidation, merger,
statutory share exchange, transfer, dissolution, liquidation or winding-up is to
take place and the time, if any such time is to be fixed, as of which the
holders of record of Common Stock (or Other Securities) shall be entitled to
exchange their shares of Common Stock (or Other Securities) for the securities
or other property deliverable upon 


                                      -9-
<PAGE>
 
such reorganization, reclassification, recapitalization, consolidation, merger,
transfer, dissolution, liquidation or winding-up. Such notice shall be mailed at
least 10 days prior to the date therein specified.

     Section 4.4.  Availability of Information. The Company will cooperate with
each holder of any Warrant, Other Security or Restricted Security in supplying
such information as may be reasonably requested by such holder to complete and
file any information reporting forms presently or hereafter required by the
Commission to report such holders beneficial ownership of Common Stock (or Other
Securities) or as a condition to the availability of an exemption from the
provisions of the Securities Act for the sale of any Restricted Securities.

     Section 4.5.  Reservation of Stock, etc. The Company will at all times
reserve and keep available, solely for issuance and delivery upon exercise of
the Warrant, the number of shares of Common Stock (or Other Securities) from
time to time issuable upon exercise of the Warrant. All shares of Common Stock
(or Other Securities) issuable upon exercise of the Warrant shall be duly
authorized and, when issued upon such exercise, shall be validly issued and, in
the case of shares, fully paid and non-assessable with no liability on the part
of the holders thereof.

                     ARTICLE V.  RESTRICTIONS ON TRANSFER

     Section 5.1.  Restrictive Legends.  Except as otherwise permitted by
this Article V, each Warrant (including each Warrant issued upon the transfer of
any Warrant) shall be stamped or otherwise imprinted with a legend in
substantially the following form:

          "This Warrant and any shares acquired upon the exercise of this
     Warrant have not been registered under the Securities Act of 1933, as
     amended, and may not be transferred, sold or otherwise disposed of in 
     the absence of such registration or an exemption therefrom under such 
     Act.  This Warrant and such shares may be transferred only in compliance
     with the conditions specified in this Warrant."

Except as otherwise permitted by this Article V, each certificate for Common
Stock (or Other Securities) issued upon the exercise of any Warrant, and each
certificate issued upon the transfer of any such Common Stock (or Other
Securities), shall be stamped or 


                                     -10-
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otherwise imprinted with a legend in substantially the following form:

          "The shares represented by this certificate have not been
     registered under the Securities Act of 1933 and may not be transferred 
     in the absence of such registration or an exemption therefrom under 
     such Act."

     Section 5.2.  Notice of Proposed Transfer; Opinions of Counsel. Prior to
any transfer of any Restricted Securities which are not registered under an
effective registration statement under the Securities Act, the holder thereof
will give written notice to the Company of such holder's intention to effect
such Transfer and to comply in all other respects with this Section 5.2. Each
such notice (a) shall describe the manner and circumstances of the proposed
Transfer and (b) shall include an opinion of legal counsel addressed to the
Company, in form and substance reasonably satisfactory to the Company, to the
effect that such Transfer may be effected without registration under the
Securities Act and any applicable state securities laws.

     Section 5.3.  Termination of Restrictions.    The restrictions imposed
by this Article V upon the transferability of Restricted Securities shall cease
and terminate as to any particular Restricted Securities when such securities
shall have been sold pursuant to an effective registration statement under the
Securities Act or otherwise become freely transferable by the holder thereof.
Upon receipt of a certificate executed by the holder of this Warrant or any
shares of Common Stock or Other Securities issued upon exercise of this Warrant,
reciting facts sufficient to establish that such holder is eligible to Transfer
such securities pursuant to Rule 144(k) promulgated under the Securities Act,
the Company shall, upon the request of such holder, issue or cause its transfer
agent to issue (in the case of the Warrant) a new Warrant or (in the case of
Common Stock or Other Securities) a new certificate free, in either case, from
any legend required by Section 5.1 hereof.

     Section 5.4.  Piggyback Registrations.

          (a) Right to Piggyback Registration.  Whenever the Company proposes to
register any of its Common Stock or Other Securities ("Common Equity
Securities") in a Qualified Registration, whether or not for sale for its own
account, the Company shall give prompt written notice (the "Piggyback Notice")
to the holders of Registrable Securities of its intention to effect such
registration.  Upon written request of any holder of 


                                     -11-
<PAGE>
 
Registrable Securities made within 10 days after delivery of any Piggyback
Notice (which request shall specify the Registrable Securities requested to be
included in such Qualified Registration by such holder), the Company shall,
subject to Sections 5.4(b) and 5.4(c), use its reasonable efforts to include in
such Qualified Registration all Registrable Securities that the holders have so
requested be included in such Qualified Registration, to permit the disposition
by such holders of such Registrable Securities; provided, however, that (1) if,
at any time after giving the Piggyback Notice and before the effective date of
the registration statement filed in connection with such Qualified Registration,
the Company determines for any reason not to register such Common Equity
Securities (other than the Registrable Securities requested to be included
therein pursuant to this Section 5.4), the Company, at its election, may give
written notice of such determination to all holders of Registrable Securities
requesting the inclusion of their Registrable Securities therein and, thereupon,
shall be relieved of its obligation to register any Registrable Securities in
connection with such registration (without prejudice, however, to the future
rights of the Holders under this Section 5.4); (2) if, at any time after giving
the Piggyback Notice and before the effective date of the registration statement
filed in connection with such Qualified Registration, the Company determines for
any reason to delay such registration of the Common Equity (other than the
Registrable Securities requested to be included therein pursuant to this Section
5.4), the Company shall be permitted to delay the registration of such
Registrable Securities for the same period as the delay in registering such
other Common Equity Securities; and (3) the Company shall not be required to
effect any registration pursuant to this Section 5.4(a) unless it shall have
received reasonable assurances that the Holders of any Registrable Securities
included therein will pay any expenses required to be paid by them as provided
in Section 5.4(d). As used herein, the term "Piggyback Registration" shall mean
any registration of Registrable Securities requested pursuant to this 
Section 5.4(a).

     (b) Priority on Piggyback Registrations.  If a Piggyback Registration
is an underwritten offering and the managing underwriter thereof advises the
Company in writing that, in its opinion, the number of shares of Registrable
Securities requested or proposed to be included in such offering exceeds the
number that can be sold in such offering without adversely affecting the
offering, including the price of the securities offered, the Company shall
include in such registration, to the extent that such may be included in such
registration without adversely affecting the offering, including the price of
the securities offered, in the opinion of such managing underwriter (1) first,
Common Equity 


                                     -12-
<PAGE>
 
Securities proposed to be sold by the Company; and (2) second, any Common Equity
Securities initially proposed to registered by the Company for the accounts of
other persons pursuant to the exercise of demand registration rights if such
securities must be included to prevent a breach of any applicable registration
rights agreement between the Company and any such person, and (3) third, such
Common Securities requested to be included in such registration pursuant to this
Warrant.

     (c) Selection of Underwriters.  If any Piggyback Registration is an
underwritten offering, the Company shall have the sole right to select the
managing underwriter or underwriters thereof.

     (d) Expenses.  The Company shall pay all expenses in connection with
piggyback registrations effected pursuant to this Section, other than the
expenses of counsel to the holder of a Warrant, and discounts, commissions and
other underwriting fees, provided, however, that if the Company is not
registering shares for sale by it, the holder of a Warrant shall pay all such
expenses of registration pro rata with the other persons registering securities
in such Piggyback Registration based upon the number of shares sold in such
Piggyback Registration.

     (e) Underwriting Agreement. If requested by the underwriters for any
underwritten offering by holders of registrable Securities, the Company will
enter into an underwriting agreement with such underwriters for such offering,
such agreement to be reasonably satisfactory in form and substance to the
Company, each such holder and the underwriters, and to contain such
representations and warranties by the Company and such other terms as are
generally prevailing and customary in agreements of this type, including,
without limitation, customary indemnifications. The holders of the Registrable
Securities will cooperate with the Company in the negotiation of the
underwriting agreement.  The holders of Registrable Securities to be distributed
by such underwriters shall be parties to such underwriting agreement.

         ARTICLE VI. OWNERSHIP, TRANSFER AND SUBSTITUTION OF WARRANTS

     Section 6.1.  Ownership of Warrants. The Company may treat the person in
whose name any Warrant is registered on the register kept at the office of the
Company maintained pursuant to subdivision (a) of Section 6.2 as the owner and
holder thereof for all purposes, notwithstanding any notice to the contrary,
except that, if and when any Warrant is properly assigned in blank, the Company
may (but shall not be obligated to) treat the bearer 


                                     -13-
<PAGE>
 
thereof as the owner of such Warrant for all purposes, notwithstanding any
notice to the contrary. Subject to Article V, a Warrant, if properly assigned,
may be exercised by a new holder without a new Warrant first having been issued.

     Section 6.2.  Office, Transfer and Exchange of Warrants.

          (a)  Office.  The Company will maintain an office where
     notices, presentations and demands in respect of this Warrant may be made
     upon it.  Such office shall be maintained at 1331 Lamar, Suite 1455,
     Houston, Texas 77010, until such time as the Company shall notify each
     holder of the Warrant of any change of location of such office.

          (b)  New Warrant.  Upon the surrender of any Warrant, properly
     endorsed, for registration of transfer or for exchange at the office of the
     Company maintained pursuant to subdivision (a) of this Section 6.2, the
     Company at its expense will (subject to compliance with Article V, if
     applicable) execute and deliver to or upon the order of the holder thereof
     a new Warrant or Warrants of like tenor, in the name of such holder or as
     such holder (upon payment by such holder of any applicable transfer taxes)
     may direct, calling in the aggregate on the face or faces thereof for the
     number of shares of Common Stock called for on the face or faces of the
     Warrant or Warrants so surrendered.

     Section 6.3.  Replacement of Warrants. Upon receipt of evidence reasonably
satisfactory to the Company of the loss, theft, destruction or mutilation of any
Warrant and, in the case of any such loss, theft or destruction of any Warrant
held by a Person other than Purchaser or any institutional investor, upon
delivery of indemnity reasonably satisfactory to the Company in form and amount
or, in the case of any such mutilation, upon surrender of such Warrant for
cancellation at the office of the Company maintained pursuant to subdivision (a)
of Section 6.2, the Company at its expense will execute and deliver, in lieu
thereof, a new Warrant of like tenor.


                                     -14-
<PAGE>
 
                           ARTICLE VII.  DEFINITIONS

     As used herein, unless the context otherwise requires, the following terms
have the following respective meanings:

     Business Day:  Any day other than a Saturday or a Sunday or a day on
which commercial banking institutions in the State of Texas are authorized by
law to be closed.  Any reference to "days" (unless Business Days are specified)
shall mean calendar days.

     Closing Price: On any date specified herein, the amount per share of the
Common Stock, equal to (a) the last sale price of such Common Stock, regular
way, on such date or, if no such sale takes place on such date, the average of
the closing bid and asked prices thereof on such date, in each case as
officially reported on the principal national securities exchange on which such
Common Stock is then listed or admitted to trading, or (b) if such Common Stock
is not then listed or admitted to trading on any national securities exchange
but is designated as a national market system security by the NASD, the last
trading price of the Common Stock on such date, or (c) if there shall have been
no trading on such date or if the Common Stock is not so designated, the average
of the closing bid and asked prices of the Common Stock on such date as shown by
the NASD automated quotation system, or (d) if such Common Stock is not then
listed or admitted to trading on any national exchange or quoted in the over-
the-counter market, the fair value thereof determined in good faith by the Board
of Directors of the Company as of a date which is within 20 days of the date as
of which the determination is to be made.

     Commission:  The Securities and Exchange Commission or any other federal
agency at the time administering the Securities Act.

     Common Stock:  As defined in the introduction to this Warrant, such term to
include (i) any stock into which such Common Stock shall have been changed or
any stock resulting from any reclassification of such Common Stock, (ii) all
other stock of any class or classes (however designated) of the Company the
holders of which have the right, without limitation as to amount, either to all
or to a share of the balance of current dividends and liquidating dividends
after the payment of dividends and distributions on any shares entitled to
preference and (iii) all stock appreciation rights, phantom stock and similar
contract rights the holders of which are entitled to payments based on or
determined by reference to the value of the Common Stock, dividends

                                     -15-
<PAGE>
 
payable with respect to Common Stock, or liquidating distributions payable with
respect to Common Stock .

     Company:  As defined in the introduction to this Warrant.

     Exchange Act:  The Securities Exchange Act of 1934, or any successor
federal statute, and the rules and regulations of the Commission thereunder, all
as the same shall be in effect at the time.

     Expiration Date:  April 9, 2002, unless terminated earlier pursuant to
Article III.

     Initial Price:  $9.90.

     Market Price:  The Market Price per share of Common Stock on any date
in question shall mean the average of the daily Closing Prices for the 10
consecutive Trading Days ending on the day before the in question.

     NASD:  The National Association of Securities Dealers, Inc.

     Other Securities:  Any stock (other than Common Stock) and other securities
of the Company or any other Person (corporate or otherwise) which the holder of
the Warrant at any time shall be entitled to receive, or shall have received
upon the exercise of the Warrant, in lieu of or in addition to Common Stock.

     Person:  Any corporation, association, partnership, joint venture, limited
liability company, trust, estate, organization, business, individual, government
or political subdivision thereof or governmental agency.

     Purchaser:  Torch Energy Advisors Incorporated

     Qualified Registration:  A registration statement of the Company under
the Securities Act on a form that permits the sale of Registrable Securities,
excluding, however, a registration statement (1) on Form S-4 or S-8 or any
successor or similar form, (2) relating to any capital stock of the Company or
options, warrants of other rights to acquire any such capital stock issued
or to be issued primarily to directors, officers or employees of the Company,
(3) filed pursuant to Rule 145 under the Securities Act or any successor or
similar provision, (4) relating to any employee benefit plan or interests
therein, (5) relating to any preferred stock or debt securities of the Company
or (6) relating to any sale of securities for other than cash.


                                     -16-
<PAGE>
 
     Registrable Securities: Common Stock or Other Securities received or
receivable upon exercise of the Warrant.

     Restricted Securities:  All of the following:  (a) any Warrant bearing
the applicable legend or legends referred to in Section 5.1, (b) any shares of
Common Stock (or Other Securities) which have been issued upon the exercise of a
Warrant and which are evidenced by a certificate or certificates bearing the
applicable legend or legends referred to in such section and (c) unless the
context otherwise requires, any shares of Common Stock (or Other Securities)
which are at the time issuable upon the exercise of a Warrant and which, when so
issued, will be evidenced by a certificate or certificates bearing the
applicable legend or legends referred to in such section.

     Securities Act:  The Securities Act of 1933, or any successor federal
statute, and the rules and regulations of the Commission thereunder, all as the
same shall be in effect at the time.

     Trading Day:  Any Monday, Tuesday, Wednesday, Thursday or Friday,
other than any day on which securities are not traded on the applicable
securities exchange or in the applicable securities market.

     Transfer:  Any sale, assignment, pledge or other disposition of any
security, or of any interest therein, which could constitute a "sale" as that
term is defined in section 2(3) of the Securities Act.

     Warrant Price: As defined in Section 2.1 of this Warrant.

                          ARTICLE VIII. MISCELLANEOUS

     Section 8.1.  No Rights or Liabilities as Stockholder. The holder of this
Warrant and all subsequent holders thereof hereby agree that except to the
extent set forth herein, no provision of this Warrant shall be construed as
conferring upon the holder hereof any rights as a stockholder of the Company or
as imposing any obligation on such holder to purchase any securities or as
imposing any liabilities on such holder as a stockholder of the Company, whether
such obligation or liabilities are asserted by the Company or by creditors of
the Company.

     Section 8.2.  Notices. All notices and other communications under this
Warrant shall be in writing and shall be mailed by registered or certified mail,
return receipt requested, addressed 


                                     -17-
<PAGE>
 
(a) if to any holder of any Warrant, to the registered address of such holder as
set forth in the register kept at the principal office of the Company, or (b) if
the Company, to the attention of its President at its office maintained pursuant
to subdivision (a) of Section 6.2, provided that the exercise of any Warrant
shall be effective in the manner provided in Article I.

     Section 8.3.  Miscellaneous.

          (a)  This Warrant may be amended, waived, discharged or terminated
     and the Company may take any action herein required to be performed by it,
     only if the Company shall have obtained the written consent to such
     amendment, action or omission to act, of the holder of this Warrant.

          (b)  THIS WARRANT SHALL BE CONSTRUED AND ENFORCED IN ACCORDANCE
     WITH THE GOVERNED BY THE LAWS OF THE STATE OF  TEXAS.

          (c)  The section headings in this Warrant are for purposes of
     convenience only and shall not constitute a part hereof.

                                       BELLWETHER EXPLORATION COMPANY

                                            /s/ J. DARBY SERE
                                       By: ____________________________
                                       Name:  J. Darby Sere
                                       Title: President


                                     -18-
<PAGE>
 
                             FORM OF SUBSCRIPTION


To ___________________________:

     The undersigned registered holder of the within Warrant  hereby 
irrevocably exercises such Warrant for, and purchases _________* shares of
Common Stock of Bellwether Exploration Company, and herewith makes payment of
$___________ therefor, and requests that the certificates for such shares be
issued in the name of, and delivered to ___________________________, whose
address is  _________________________________________.

Dated:                                      (Signature must conform in all
                                             respects to name of holder as
                                             specified on the face of Warrant)


                                            __________________________________
                                                     (Street Address)


                                            __________________________________
                                            (City)      (State)     (Zip Code)


_______________________________

          *Insert the number of shares called for on the face of this Warrant
     (or, in the case of a partial exercise, the portion thereof as to which
     this Warrant is being exercised), in either case without making any
     adjustment for additional shares of Common Stock or any other stock or
     other securities or property or cash which, pursuant to the adjustment
     provisions of this Warrant, may be delivered upon exercise. In the case of
     a partial exercise, a new Warrant or Warrants will be issued and delivered,
     representing the unexercised portion of the Warrant, to the holder
     surrendering the Warrant.


                                     -19-
<PAGE>
 
                              FORM OF ASSIGNMENT

                [To be executed only upon Transfer of Warrant]


     For value received, the undersigned registered holder of the within
Warrant hereby sells, assigns and transfers unto  _________________ the right
represented by such Warrant to purchase shares of Common Stock of Bellwether
Exploration Company to which such Warrant relates, and appoints
_________________________________ as its duly authorized and empowered attorney-
in-fact to make such transfer on the books of maintained for such purpose, with
full power of substitution in the premises.

Dated:                                      (Signature must conform in all
                                             respects to name of holder as
                                             specified on the face of Warrant)


                                            __________________________________
                                                     (Street Address)


                                            __________________________________
                                            (City)      (State)     (Zip Code)

Signed in the presence of:


_______________________________


                                     -20-