<PAGE>   1


                                                                 EXHIBIT 10.10.1

                 PROMISSORY NOTE AND WARRANT PURCHASE AGREEMENT


         THIS PROMISSORY NOTE AND WARRANT PURCHASE AGREEMENT ("Agreement") is
made as of the 22nd day of May, 1998, by and among Odyssey HealthCare, Inc., a
Delaware corporation (the "Company"), those persons whose names are set forth on
Exhibit A hereto (collectively referred to herein as "Founders"), and the
Purchasers identified on Exhibit B attached hereto (collectively referred to
herein as "Purchasers" and individually as a "Purchaser").

                                    RECITALS:

         WHEREAS, the Purchasers desire to purchase, and the Company desires to
issue, convertible promissory notes (the "Notes") with an aggregate principal
amount of One Million Five Hundred Thousand Dollars ($1,500,000);

         WHEREAS, in consideration of the purchase by the Purchasers of the
Notes, the Company desires to sell and issue warrants (the "Warrants") to
purchase that number of shares of the Company's Series B Convertible Preferred
Stock ("Series B Stock") as determined in this Agreement and in the Warrants;

         WHEREAS, the Company, the Founders and certain of the Purchasers are
parties to an Amended and Restated Registration Rights Agreement, dated as of
February 12, 1997 (the "Rights Agreement"), which grants certain registration
and other rights to the parties thereof, and to an Amended and Restated
Stockholders' Agreement, dated as of February 12, 1997 (the "Stockholders'
Agreement"), which grants certain information and other rights to the parties
thereof,

         WHEREAS, the Company, the Founders and the Purchasers holding (i) at
least a majority of the outstanding shares of Restricted Stock (as such term is
defined in the Rights Agreement) wish to amend the Rights Agreement to include
as Preferred Shares (as such term is defined in the Rights Agreement) the Series
B Stock issuable upon exercise of the Warrants (the "Warrant Shares"), and (ii)
at least a majority of Shares (as such term is defined in the Stockholders'
Agreement) wish to amend the Stockholders' Agreement to include as Preferred
Shares (as such term is defined in the Stockholders' Agreement) the Warrant
Shares; and

         WHEREAS, the parties also wish to set forth certain representations,
warranties, covenants, and agreements relating to the purchase of the Notes and
Warrants provided for herein.

         NOW, THEREFORE, THE PARTIES HEREBY AGREE AS FOLLOWS:

         1. Authorization and Issuance of Notes and Warrants.

                  1.1 Authorization. The Company has authorized the sale and
issuance of the Notes and Warrants to the Purchasers.



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                  1.2 Sale and Issuance of Notes. Subject to the terms and
conditions hereof, the Company agrees to sell and issue to each of the
Purchasers, and the Purchasers severally agree to purchase from the Company, a
Note in the form attached as Exhibit C hereto, in the amount set forth opposite
such Purchaser's name on Exhibit B hereto.

                  1.3 Sale and Issuance of Warrants. Subject to the terms and
conditions hereof, the Company agrees to sell and issue to each of the
Purchasers, and the Purchasers severally agree to purchase from the Company,
Warrants originally exercisable, at an exercise price of $1.25 per share, for
that number of shares of Series B Stock set forth opposite such Purchaser's name
on Exhibit B hereto, in the form attached hereto as Exhibit D at a purchase
price of $.025 for each share of Series B Stock for which such Warrants are
originally exercisable, which Warrants and Warrant Shares shall be subject to
adjustment, as described in the Warrants.

         2. Closing.

                  2.1 Closing; Closing Date. The closing of the issuance of the
Notes and Warrants under this Agreement (the "Closing") shall take place on the
date of this Agreement (the "Closing Date"), in accordance with arrangements
mutually satisfactory to the Purchasers and counsel for the Company.

                  2.2 Closing Delivery. At the Closing, upon delivery to the
Company by wire transfer or check made payable to the order of the Company of
the aggregate purchase price for the Note and the Warrant set forth opposite
such Purchaser's name on Exhibit B hereto, the Company will deliver to each
Purchaser (a) a Note payable to the Purchaser in the principal amount set forth
opposite such Purchaser's name on Exhibit B hereto and (b) a Warrant to purchase
that number of shares of Series B Stock set forth opposite such Purchaser's name
on Exhibit B hereto.

         3. Amendments to the Rights Agreement. Each Purchaser agrees to be
bound by the terms and conditions of the Rights Agreement. In light of the
foregoing and in order to grant the Purchasers the registration rights and other
rights set forth in the Rights Agreement in respect of the Warrant Shares, the
Rights Agreement shall be amended as follows:

                           (a) The term "Preferred Shares" as set forth in
Section 1 of the Rights Agreement shall include the Warrant Shares hereunder.

         4. Amendments to the Stockholders' Agreement. Each Purchaser agrees to
be bound by the terms and conditions of the Stockholders' Agreement. In light of
the foregoing and in order to grant the Purchasers the information rights and
other rights set forth in the Stockholders' Agreement in respect of the Warrant
Shares, the Stockholders' Agreement shall be amended as follows:

                           (a) The term "Preferred Shares" as set forth in the
recitals of the Stockholders' Agreement shall include the Warrant Shares
hereunder.

                           (b) The issuance of the Notes, Warrants and Warrant
Shares shall be excluded from the right of first refusal in Section 2 of the
Stockholders' Agreement.




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<PAGE>   3

         5. Amendment to Certificate of Incorporation. In order to authorize a
sufficient number of shares of Series B Stock for issuance upon exercise of the
Warrants, the Company shall adopt and file with the Secretary of State of the
State of Delaware on or before the Closing an amendment to its Second Amended
and Restated Certificate of Incorporation (the "Certificate of Incorporation")
substantially in the form attached hereto as Exhibit E to authorize an
additional _____________ shares of Series B Stock.

         6. Representations And Warranties of the Company.

                  6.1 Authorization. All corporate action on the part of the
Company, its officers, directors, and shareholders necessary for the
authorization, execution, and delivery of this Agreement, the performance of all
the Company's obligations hereunder and for the authorization, issuance, sale,
and delivery of the Notes, the Warrants, and the Warrant Shares has been taken
or will be taken prior to the Closing. This Agreement, the Notes and the
Warrants when executed and delivered, shall constitute the valid and legally
binding obligations of the Company enforceable in accordance with their
respective terms, subject to the laws of general application relating to
bankruptcy, insolvency, and the relief of debtors.

                  6.2 Validity of Notes, Warrants and Warrant Shares. The Notes
and Warrants, when issued in accordance with the terms of this Agreement, shall
be duty and validly issued. The issuance of the Warrants and any subsequent
issuance of the Warrant Shares are not and will not be subject to any rights of
first refusal or preemptive rights and, when issued, sold, and delivered in
compliance with the provisions of this Agreement and the terms of the Warrants
and in accordance with the Company's Certificate of Incorporation, as amended,
the Warrants and the Warrant Shares will be validly issued, fully paid, and
nonassessable, and will be free of any liens or encumbrances; provided, however,
that the Warrants and the Warrant Shares may be subject to restrictions on
transfer under state and/or federal securities laws as set forth herein or as
otherwise required by such laws at the time a transfer is proposed.

                  6.3 Governmental Consents. All consents, approvals, orders, or
authorizations of, or registrations, qualifications, designations, declarations,
or filings with, any governmental authority, required on the part of the Company
in connection with the valid execution and delivery of this Agreement, the
offer, sale, or issuance of the Notes, the Warrants, and the Warrant Shares, or
the consummation of any other transaction contemplated hereby shall have been
obtained and will be effective at the Closing, except for notices required or
permitted to be filed with certain state and federal securities commissions,
which notices will be filed on a timely basis.

         7. Representations and Warranties of the Purchasers. Each Purchaser,
severally and not jointly, hereby represents and warrants to the Company as
follows:

                  7.1 Legal Authority. It has the requisite legal power to enter
into this Agreement, to purchase the Note and Warrants hereunder and to carry
out and perform its obligations under the terms of this Agreement.




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                  7.2 Due Execution. This Agreement has been duly authorized,
executed, and delivered by it, and, upon execution and delivery by the Company,
this Agreement will be a valid and binding agreement of it.

                  7.3 Investment Representations.

                           (a) It is acquiring the Note and Warrants for its own
account, not as nominee or agent, for investment and not with a view to, or for
resale in connection with, any distribution or public offering of the Note,
Warrants, or Warrant Shares within the meaning of the Securities Act of 1933, as
amended (the "1933 Act").

                           (b) It understands that (i) the Note, Warrants, and
Warrant Shares have not been registered under the 1933 Act by reason of a
specific exemption therefrom, that they must be held by it indefinitely, and
that Purchaser must, therefore, bear the economic risk of such investment
indefinitely, unless a subsequent disposition thereof is registered under the
1933 Act or is exempt from such registration; and (ii) the Note, Warrants, and
each certificate representing the Warrant Shares will be endorsed with the
following legend:

         "THESE SECURITIES HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF
         1933, AS AMENDED. THEY MAY NOT BE SOLD, OFFERED FOR SALE, PLEDGED,
         HYPOTHECATED OR OTHERWISE TRANSFERRED EXCEPT PURSUANT TO AN EFFECTIVE
         REGISTRATION STATEMENT UNDER SUCH ACT OR PURSUANT TO AN OPINION OF
         COUNSEL SATISFACTORY TO THE COMPANY THAT SUCH REGISTRATION IS NOT
         REQUIRED, OR UNLESS SOLD PURSUANT TO RULE 144 UNDER SUCH ACT."

                           (c) It has been furnished with such materials and has
been given access to such information relating to the Company as it or its
qualified representative has requested and it has been afforded the opportunity
to ask questions regarding the Company, the Notes, and the Warrants, all as it
has found necessary to make an informed investment decision.

                           (d) By reason of its business or financial
experience, or the business or financial experience of its professional advisor,
it has the capacity to protect its own interests in connection with this
transaction.

                           (e) If it is a corporation, partnership, trust, or
other entity, it was not formed for the specific purpose of acquiring the Notes
or Warrants offered hereunder.

                           (f) It is an "accredited investor" as provided under
the 1933 Act and regulations adopted thereunder; and all information supplied by
such Purchaser to the Company with respect to his or its purchase of the Notes
and Warrants has been and shall be true, complete, and accurate.




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         8. Miscellaneous.

                  8.1 Governing Law. This Agreement shall be governed by and
construed under the laws of the State of Texas as applied to agreements among
Texas residents, made and to be performed entirely within the State of Texas.

                  8.2 Successors and Assigns. Except as otherwise expressly
provided herein, the terms and conditions of this Agreement shall inure to the
benefit of, and be binding upon, the successors, assigns, heirs, executors, and
administrators of the parties hereto.

                  8.3 Entire Agreement. This Agreement, the Exhibits hereto, and
the other documents delivered pursuant hereto constitute the full and entire
understanding and agreement among the parties with regard to the subjects hereof
and no party shall be liable or bound to any other party in any manner by any
representations, warranties, covenants, or agreements except as specifically set
forth herein or therein.

                  8.4 Separability. Any invalidity, illegality, or limitation of
the enforceability with respect to any Purchaser of any one or more of the
provisions of this Agreement, or any part thereof, whether arising by reason of
the law of any such Purchaser's domicile or otherwise, shall in no way affect or
impair the validity, legality, or enforceability of this Agreement with respect
to other Purchasers. In case any provision of this Agreement shall be invalid,
illegal, or unenforceable, the validity, legality, and enforceability of the
remaining provisions shall not in any way be affected or impaired thereby.

                  8.5 Amendment and Waiver. Any term of this Agreement may be
amended and the observance of any term of this Agreement may be waived with the
written consent of the Company and the Purchasers of more than 50% of the
outstanding principal amount of the Notes.

                  8.6 Communications. All notices or other communications
hereunder shall be in writing and shall be given by personal delivery,
facsimile, overnight courier service, or by registered or certified mail
(postage prepaid and return receipt requested) addressed as set forth below (or
at such other address as a party may designate by notice to the other parties):

                  If to the Company:

                  ODYSSEY HEALTHCARE, INC.
                  717 North Harwood Drive
                  Suite 2580
                  Dallas, Texas 75201

                  If to the Purchasers:

                  At the respective addresses indicated for such Purchasers on
                  the signature pages hereof

Notice sent pursuant to or required by this Agreement shall be deemed given (i)
in the case of personal delivery, on the date of such delivery, (ii) in the case
of telex or facsimile transmission, on the date on which the sender receives
confirmation by telex or facsimile transmission that




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such notice was received by the addressee, provided that a copy of such
transmission is additionally sent by overnight air courier or mail as set forth
in (iii) or (iv), respectively, below; (iii) in the case of overnight air
courier, on the next business day following the day sent, with receipt confirmed
by the courier; and (iv) in the case of mailing by first class certified or
registered mail, postage prepaid, return receipt requested, on the fifth
business day following such mailing.

                  8.7 Expenses. The Company shall bear all expenses and legal
fees incurred by it and by the Purchasers with respect to this Agreement and the
transactions contemplated hereby; provided, however, that the Purchasers' legal
fees shall be fixed at Seven Thousand Dollars ($7,000).

                  8.8 Titles and Subtitles. The titles of the paragraphs and
subparagraphs of this Agreement are for convenience of reference only and are
not to be considered in construing this Agreement.

                  8.9 Counterparts. This Agreement may be executed in any number
of counterparts, each of which shall be deemed an original, but all of which
together shall constitute one instrument.

                  8.10 Miscellaneous. Each Purchaser hereby agrees that, if so
requested by the Company and the underwriter, such Purchaser shall not publicly
sell any securities issued upon exercise of the Warrants or conversion of the
Notes (other than securities being registered in such offering) without the
consent of such underwriters, until one hundred eighty (180) days following the
effective date of the registration statement relating to the Company's initial
underwritten public offering, provided that said one hundred eighty (180) day
period may be extended for up to an additional sixty (60) days upon the majority
vote of both the Company's Board of Directors and the Company's shareholders;
provided, however, all persons entitled to registration rights with respect to
shares of Common Stock who are not parties to the Rights' Agreement, all other
persons selling shares of Common Stock in such offering and all executive
officers and directors of the Company shall also have agreed not to sell
publicly their Common Stock under circumstances pursuant to the terms set forth
in this Section 8.10.




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         IN WITNESS WHEREOF, the Company has caused this Promissory Note and
Warrant Purchase Agreement to be signed by its duly authorized officer.

                                          ODYSSEY HEALTHCARE, INC.


                                          By:   /s/ Bradley J. Velie
                                             -----------------------------------

                                          Name:     Bradley J. Velie
                                               ---------------------------------

                                          Title:    VP
                                                --------------------------------

                                          PURCHASER:



                                          --------------------------------------
                                          (Please insert Purchaser name above)


                                          By:
                                             -----------------------------------
                                                        (sign here)


                                          Title:
                                                --------------------------------

<PAGE>   8



                                          FOUNDERS:



                                          --------------------------------------
                                                  Richard C. Burnham



                                          --------------------------------------
                                                  David C. Gasmire



                                          --------------------------------------
                                                  David W. Cross



                                          --------------------------------------
                                                  David L. Steffy



                                          --------------------------------------
                                                  Robert H. Calland



<PAGE>   9



                                          THREE ARCH PARTNERS, L.P.

                                          By: Three Arch Management, L.P.
                                          Its: General Partner


                                          --------------------------------------


                                          THREE ARCH ASSOCIATES, L.P.

                                          By: Three Arch Management, L.P.
                                          Its: General Partner


                                          --------------------------------------




<PAGE>   10

                                    EXHIBIT A

                              SCHEDULE OF FOUNDERS


Richard C. Burnham

David C. Gasmire

David W. Cross

David L. Steffy

Robert H. Calland


                                      E-1
<PAGE>   11

                                    EXHIBIT B

                             SCHEDULE OF PURCHASERS

<TABLE>
<CAPTION>

                                                NUMBER OF SHARES
                                                   OF SERIES B         PURCHASE
                                                   CONVERTIBLE         PRICE OF
                                                 PREFERRED STOCK      WARRANT AT
                                    AMOUNT      EXERCISABLE UNDER     $0.025 PER         TOTAL
             NAME                   OF NOTE          WARRANTS            SHARE         INVESTMENT
-----------------------------    -------------  -----------------    -------------    -------------

<S>                              <C>                     <C>       <C>              <C>
Three Arch Partners, L.P.        $  318,910.21           25,512    $      637.80    $  319,548.01
Three Arch Associates, L.P.          71,746.06            5,739           143.48        71,889.54
Weiss, Peck & Greer Venture
     Associates III, L.P.           175,144.05           14,011           350.28       175,494.33
WPG Enterprise Fund II, L.P.        210,635.79           16,850           421.25       211,057.04
Oak Investment Partners VI          371,593.11           29,727           743.18       372,336.29
Oak VI Affiliates                     8,669.97              693            17.33         8,687.30
Highland Capital Partners III
     Limited Partnership            166,448.83           13,315           332.88       166,781.71
Highland Entrepreneurs' Fund
     III Limited Partnership          6,935.37              554            13.85         6,949.22
Collinson Howe Venture
     Partners, Inc.                  10,836.51              866            21.65        10,858.16
David L. Steffy                     159,080.10           12,726           318.15       159,398.26
                                 -------------    -------------    -------------    -------------
                                 $1,500,000.00          119,993    $    2,999.85     1,502,999.85
                                 =============    =============    =============    =============
</TABLE>