<PAGE>   1
                                                                  EXHIBIT 10.5.2


                             FIRST AMENDMENT TO THE
                            ODYSSEY HEALTHCARE, INC.
                                STOCK OPTION PLAN


         THIS FIRST AMENDMENT is effective as set forth below and is made by
Odyssey Healthcare, Inc., a Delaware corporation (the "Company").

                                   WITNESSETH:

         WHEREAS, the Company sponsors the Odyssey Healthcare, Inc. Stock Option
Plan (the "Plan");

         WHEREAS, the Board of Directors of the Company (the "Board") maintains
the right to unilaterally amend the Plan in certain respects and to amend the
Plan in certain respects subject to stockholder approval;

         WHEREAS, the Board's right to amend the Plan includes the right to
modify the group of non-employees eligible to receive options which are not
designed to comply with section 422 of the Internal Revenue Code of 1986, as
amended (the "Code") (such options herein referred to as "Nonqualified Incentive
Stock Options");

         WHEREAS, the Board's right to amend the Plan, subject to stockholder
approval, includes the right to increase the number of shares of the Company
common stock par value $0.001 (the "Common Stock"), of the Company that may be
subject to options under the Plan;

         WHEREAS, the Board's right to amend the Plan further includes the right
to remove transfer restrictions applicable to Nonqualified Incentive Stock
Options;

         WHEREAS, the Board desires to clarify that the Plan allows for the
grant of Nonqualified Incentive Stock Options to non-employee directors of the
Company;

         WHEREAS, the Board desires to increase the number of shares of Common
Stock that may be issued and sold under the Plan; and

         WHEREAS, the Board desires to permit the transfer of Nonqualified
Incentive Stock Options in limited circumstances.

         NOW, THEREFORE, the Plan is hereby amended as follows:

         1. Section 3.0 of the Plan is hereby amended and restated to read in
its entirety as follows, effective as of February 22, 1996:

                  3.0 ELIGIBILITY FOR PARTICIPATION. Subject to the terms and
provisions of the Plan, participants in the Plan shall be selected by the
Committee from the executive officers (whether or not they are directors),
certain other key employees of the Company and non-employee directors of the
Company (the "Participants"). In making this selection and in determining the
form and amount of options to be granted, the Committee shall







                                       1
<PAGE>   2

consider any factors deemed relevant, including the individual's functions,
responsibilities, value of services to the Company, and past and potential
contributions to the Company's profitability and sound growth. Participants
selected by the Committee shall be employees or non-employee directors of the
Company. A Participant may hold more than one option, but only on the terms and
subject to the restrictions hereafter set forth. No person shall be eligible to
receive an option for a larger number of shares than is recommended for him or
her by the Committee. Notwithstanding the foregoing, a non-employee director of
the Company shall only be entitled to receive Nonqualified Incentive Stock
Options granted pursuant to Section 25.

         2. Section 4.0 is hereby amended and restated to read in its entirety
as follows, effective as of January 18, 2001:

                  4.0 SHARES SUBJECT TO THE PLAN. The shares of stock which may
be subject to options under the Plan shall be shares of the Company's common
stock, either authorized and unissued shares or shares issued and held in its
treasury. The stock subject to the options shall be Two Million Five Hundred and
Seventy-Eight Thousand (2,578,000) shares of Common Stock, $0.001 par value,
hereinafter referred to as "Common Stock."

         3. Section 5.0(E) is hereby amended and restated to read in its
entirety as follows, effective as of February 22, 1996:

                  (E) RESTRICTIONS ON TRANSFER OF OPTION: Each Incentive Stock
Option Agreement shall state that the option is not transferable by the Optionee
except by will or by the laws of descent and distribution. Incentive Stock
Options may be exercised during the lifetime of the Optionee only by the
Optionee, and after the death of the Optionee, only as provided in Section 7.0.
Nonqualified Incentive Stock Options shall be transferable as set forth herein:

                           (1) The Committee may, in its discretion, permit a
Participant to transfer all or any portion of a Nonqualified Incentive Stock
Option granted pursuant to Section 25, or authorize all or a portion of such
option to be granted to a Participant to be on terms which permit transfer by
such Participant; provided that, in either case the transferee or transferees
must be an individual or individuals approved by the Committee ("Permitted
Transferees"); provided further that, subsequent transfers of such Nonqualified
Incentive Stock Option transferred as provided above shall be prohibited except
for subsequent transfers back to the original holder of the Nonqualified
Incentive Stock Option and transfers to other Permitted Transferees of the
original holder. Agreements evidencing the Nonqualified Incentive Stock Option
with respect to which such transferability is authorized at the time of grant
must be approved by the Committee, and must expressly provide for
transferability in a manner consistent with this Section 5.0(E).

                           (2) A Nonqualified Incentive Stock Option may be
transferred, to a Permitted Transferee, pursuant to a domestic relations order
entered or approved by a court of competent jurisdiction upon delivery to the
Company of written notice of such transfer and a certified copy of such order.

                           (3) Except as expressly permitted in this Section
5.0(E), Nonqualified Incentive Stock Options shall not be transferable other
than by will or the laws of descent and distribution.




                                       2
<PAGE>   3

                           (4) Following the transfer of any Nonqualified
Incentive Stock Option as contemplated by this Section 5.0(E), such Nonqualified
Incentive Stock Option shall continue to be subject to the same terms and
conditions as were applicable immediately prior to transfer, provided that the
term "Participant" shall be deemed to refer to the Permitted Transferee, the
recipient under a qualified domestic relations order, the estate or heirs of a
deceased Participant, or other transferee, as applicable, to the extent
appropriate to enable the Participant to exercise the transferred Nonqualified
Incentive Stock Option in accordance with the terms of this Plan and applicable
law. The provisions of the Nonqualified Incentive Stock Option relating to
exercisability thereof shall continue to be applied with respect to the original
Participant and, following the occurrence of any such events described therein
the Nonqualified Incentive Stock Option shall be exercisable by the Permitted
Transferee, the recipient under a qualified domestic relations order, the estate
or heirs of a deceased Participant, or other transferee, as applicable, only to
the extent and for the periods that would have been applicable in the absence of
the transfer.

                           (5) Any Participant desiring to transfer a
Nonqualified Incentive Stock Option pursuant to this Section 5.0(E) shall make
application therefor in the manner and time specified by the Committee and shall
comply with such other requirements as the Committee may require to assure
compliance with all applicable securities laws. The Committee shall not give
permission for such a transfer if it would give rise to short-swing liability
under Section 16(b) of the Securities Exchange Act of 1934, as amended, or if it
may not be made in compliance with all applicable federal, state and foreign
securities laws.

         4. Section 28 is hereby added to the Plan to read as follows:

                  28.0 RIGHT OF FIRST REFUSAL. If any Participant or any
transferee (collectively, the "Transferor"), regardless of whether such
Participant is the original holder of the Option contemplated in this Section
28.0, proposes to sell, transfer, assign, hypothecate, make gifts of or in any
manner dispose of, encumber, or alienate (each individually constituting a
"Transfer") to a transferee, other than a Permitted Transferee, of any Option
issued under this Plan, or Common Stock obtained through the exercise or vesting
of any Option, held by such Transferor, either pursuant to a bona fide offer
("Offer") from a potential transferee ("Offeror") or by effecting a gift of the
Option or Common Stock without consideration ("Gift") to a donee ("Donee"), then
the Transferor must comply with the provisions of this Section 28.0, including
without limitation acknowledging and allowing the applicable time periods to
lapse with respect to the rights of the Company as provided herein, before
accepting any such Offer or otherwise effecting the Transfer of any Common Stock
pursuant to such Offer, or effecting any such Gift.

                           (A) STATEMENT OF OFFER. Before accepting any Offer or
effecting any Gift, the Transferor shall obtain from the Offeror or Donee, as
the case may be, a statement ("Statement") in writing addressed to the
Transferor and signed by the Offeror or Donee, setting forth: (1) the date of
the Statement (the "Statement Date"); (2) the number of shares of Stock covered
by the Offer or Gift and, in the case of an Offer, the price per share to be
paid by the Offeror and the terms of payment of such price; (3) the Offeror's or
Donee's willingness to be bound by the terms of this Section 28.0 and execute
and deliver to the Company such documentation as required under this Section
28.0; (4) the Offeror's or Donee's name, address and telephone number; and (5)
the Offeror's or Donee's willingness to supply any additional





                                       3
<PAGE>   4

information about himself as may be reasonably requested by the Company.
Promptly upon receipt of a Statement, and before accepting the Offer or
effecting the Gift to which the Statement relates, the Transferor shall deliver
to the Company (a) a copy of the Statement, and (b) in the case of an Offer,
evidence reasonably satisfactory to the Company as to the Offeror's financial
ability to consummate the proposed purchase.

                           (B) COMPANY RIGHTS. Subject to the provisions of
Section 28.0(A), upon receipt of a copy of the Statement, the Company shall have
the exclusive right and option (the "Right"), but not the obligation, to
purchase all of the shares of Common Stock that the Offeror proposes to purchase
from the Transferor or, in the case of a Gift, that the Transferor proposes to
give to the Donee (collectively, "Subject Securities") (1) in the case of an
Offer, for the per share price and on the terms as set forth in the Statement;
provided, however, that if the purchase price is payable in whole or in part in
property (which term shall include the securities of any issuer other than the
Company) other than cash, the Company may pay, in lieu of such property, a sum
of cash equal to the fair market value of such property as determined by the
Transferor and the Company in good faith or, if the Transferor and the Company
do not agree on the fair market value of such property within five days after
the Company delivers written notice (as described below) of its intention to
exercise the Right, then the Transferor and the Company shall select one
independent appraiser (with each of the Transferor and the Company jointly
bearing one-half of the expense of the appraiser) to determine the fair market
value of that property and the appraised fair market value of that property as
determined by such appraiser shall be deemed the fair market value of that
property for purposes of this Section 28.0, or (2) in the case of a Gift, the
fair market value of the Subject Securities, as determined in good faith by the
Board of Directors; provided that the Transferor may elect to retain the Subject
Securities rather than sell the Subject Securities at the fair market value as
determined by the Board of Directors by giving written notice thereof to the
Company within five days after such determination by the Board of Directors is
received in writing by the Transferor. The Company shall exercise the Right by
giving written notice thereof to the Transferor. Upon exercising the Right, the
Company shall have the obligation, to the extent it lawfully may do so, to
purchase the Subject Securities within 30 days after the date of the Company's
receipt of its copy of the Statement on and subject to the terms and conditions
hereof. Failure by the Company to exercise the Right, or failure by the Company
to otherwise perform its obligations under this Section 28.0, within the 30-day
period herein prescribed shall be deemed an election by the Company not to
exercise the Right. If the Company exercises the Right and is unable for any
reason to perform its obligations thereunder in accordance with this Section
28.0, the Company may assign all or a portion of its rights under the Right to
any one or more of the Company's stockholders (other than the Transferor)
("Assignee Stockholder"), as the Board of Directors shall determine, in its sole
and absolute discretion.

                           (C) PURCHASE OF LESS THAN ALL SHARES. Anything in
this Section 28.0 to the contrary notwithstanding, the Company and any Assignee
Stockholder individually may, pursuant to the exercise of the Right, purchase
fewer than all of the Subject Securities provided that such persons in the
aggregate purchase all, and not less than all, of the Subject Securities, and it
shall be a condition precedent to the obligation of any of such persons to
purchase any Subject Securities, and to the obligation of the Transferor to sell
any Subject Securities, that all, and not less than all, of the Subject
Securities have been elected to be purchased pursuant to the exercise of the
Right.




                                       4
<PAGE>   5

                           (D) FAILURE TO EXERCISE RIGHT OR CONSUMMATE
TRANSACTION. If the Company elects not to exercise the Right, or if the Right is
exercised and the obligations to be performed thereunder by the Company are not
performed in accordance with this Section 28.0, or if the Company's rights are
assigned to an Assignee Stockholder and such Assignee Stockholder fails to
perform his or its obligations under the assigned Right in accordance with this
Section 28.0, then, subject to the application of any applicable state or
federal securities laws, the Transferor may dispose of all of the Subject
Securities within 90 days after the date of the Statement at the per share price
and on the terms, if any, as set forth in the Statement free and clear of the
terms of this Section 28.0; provided, however, that (1) any subsequent transfer
by the Offeror or Donee, as applicable, shall once again be subject to this
Section 28.0 and (2) if the sale or gift of the Subject Securities is not
consummated within such 90-day period, then the Transfer of any such Stock shall
once again be subject to the terms of this Section 28.0.

                           (E) APPLICATION AND EXPIRATION. The provisions of
this Section 28.0, which pertain to rights of first refusal, shall only apply
upon the expiration of the Second Amended and Restated Stockholders' Agreement
dated June 30, 1998, or any successor agreement. The rights and obligations
pursuant to this Section 28.0 hereof will terminate upon the date on which the
initial registration of the Common Stock under Section 12(b) or 12(g) of the
Securities Exchange Act of 1934, as amended, first becomes effective.

         NOW, THEREFORE, be it provided that, except as provided above, the Plan
shall continue to read in its current state.

         IN WITNESS WHEREOF, this First Amendment has been executed by a duly
authorized officer of the Company as of the date specified below and effective
as set forth herein.

                            ODYSSEY HEALTHCARE, INC.
                            a Delaware corporation

                            By: /s/ RICHARD R. BURNHAM
                               -----------------------------------------------
                               Richard R. Burnham, President, Chief Executive
                               Officer and Chairman of the Board


                            Date: January 31, 2001
                                 ---------------------------------------------




                                       5