<PAGE>   1
                                                                     EXHIBIT 2.3



                  SECOND AMENDMENT TO STOCK PURCHASE AGREEMENT


         SECOND AMENDMENT (the "Second Amendment") dated as of September 30, 
1998 by and among Lamar Advertising Company, a Delaware corporation ("Buyer"),
Outdoor Communications, Inc., a Delaware corporation (the "Company"), and each
of the stockholders of the Company (the "Stockholders").

         WHEREAS Buyer, the Company and each of the Stockholders are parties to
that certain Stock Purchase Agreement dated as of August 10, 1998 as amended by
the First Amendment to Stock Purchase Agreement dated as of August 24, 1998
(the "Agreement"), pursuant to which each of the Stockholders has agreed to
sell to Buyer, and Buyer has agreed to purchase, all of the Company Securities
held by such Stockholder; and

         WHEREAS Buyer, the Company and the Stockholders desire to amend the
Agreement to provide for payment for the Company Securities held by certain
Stockholders to be paid by the issuance of promissory notes by Buyer.

         NOW, THEREFORE, for good and valuable consideration, the undersigned
hereby agree as follows:

         1.   Capitalized terms not otherwise defined herein shall have the
meanings ascribed to such terms in the Agreement.

         2.   The Agreement is hereby amended, as of the effective date of
this Second Amendment, by inserting the following language after the phrase "in
Section 2.2" in the first line of Section 2.4(b):

         ", less amounts payable to each of John C Stanley IV, A.B. Isbell and
         Ricky W. Thomas through the issuance of promissory notes in respect of
         the Company Securities owned by them in accordance with the last
         sentence of this subsection (b),"

         3.   The Agreement is hereby amended, as of the effective date of
this Second Amendment, by adding the following sentence to Section 2.4(b):

         "Notwithstanding anything to the contrary in this Agreement, Buyer
         shall deliver to each of John C Stanley IV, A.B. Isbell and Ricky W.
         Thomas, as payment for the Company Securities owned by each such
         individual, a promissory note, in the form attached hereto as Exhibit
         B, in a principal amount equal to the amount set forth opposite such
         person's name in Schedule 2.4 attached hereto."

         4.   Exhibit A to the Agreement is hereby amended, as of the
effective date of this Second Amendment, by deleting the number "1,030.31" set
forth in the Shares of Series A Preferred Stock column opposite the name of
Norman Isbell and substituting therefor ",".

         5.   Buyer, the Company and the Stockholders hereby agree that
there shall be no adjustment to the purchase price payable to Stockholders for
capital expenditures and acquisitions by the Company between August 10, 1998
and the Closing Date.




                                      1
<PAGE>   2



         6.   Section 7.5 is amended by adding at the end thereof the following:

              ", except that the Stockholder Note will be paid by issuance of a
              promissory note in the form attached hereto as Exhibit B in the
              principal amount of $2,000,000."

         7.   The effective date of this Second Amendment shall be the date
first set forth above.

         8.   As amended by this Second Amendment, the Agreement is in all
respects ratified and confirmed, and as so amended by this Second Amendment,
the Agreement shall be read, taken and construed as one and the same
instrument.

         9.   This Second Amendment may be executed in any number of 
counterparts and by the parties hereto in separate counterparts, each of which
so executed shall be deemed to be an original, but all of such counterparts
shall together constitute but one and the same instrument.

         10.  This Second Amendment shall be governed in accordance with the
laws of the State of Delaware without regard to principles of conflicts of law.

                                [END OF TEXT]





                                       2
<PAGE>   3



         IN WITNESS WHEREOF the parties hereto have caused this Second
Amendment to be executed as of the date set forth above by their duly
authorized representatives.


                                 BUYER:
                                 
                                 LAMAR ADVERTISING COMPANY
                                 
                                 
                                 By:     /s/ Keith A. Istre                 
                                      -----------------------------------------
                                      Keith A. Istre, Chief Financial Officer
                                 
                                 
                                 
                                 COMPANY:
                                 
                                 OUTDOOR COMMUNICATIONS, INC.
                                 
                                 
                                 By:     /s/ John C Stanley IV             
                                      -----------------------------------------
                                      John C Stanley IV, Chairman
                                 
                                 
                                 
                                 STOCKHOLDERS:
                                 
                                 
                                 By:     /s/ Stephen F. Gormley            
                                      -----------------------------------------
                                      Stephen F. Gormley, on behalf of each of 
                                      the Stockholders, on behalf of Media/
                                      Communications Partners II Limited 
                                      Partnership in its capacity as a
                                      Stockholders' Representative
                                 
                                 
                                 
                                 By:     /s/ John C Stanley IV             
                                      -----------------------------------------
                                      John C Stanley IV, on behalf of each of 
                                      the Stockholders, in his capacity as a 
                                      Stockholders' Representative





                                       3
<PAGE>   4





                                  SCHEDULE 2.4



<TABLE>
<CAPTION>
----------------------------------------------------------------------------------------
                     Name                       Amount to be Paid Through Issuance 
                                                of a Note
----------------------------------------------------------------------------------------
  <S>                                          <C>
  John C Stanley IV                                      $22,803,673.00
----------------------------------------------------------------------------------------
  A.B. Isbell                                             19,007,475.00
----------------------------------------------------------------------------------------
  Ricky W. Thomas                                          1,196,178.00
----------------------------------------------------------------------------------------
</TABLE>





                                       4