While AS has plans to open a lounge at SAN that is still in the future. As it is the Aspire lounge is only a contract lounge that provides access to those having
(1) having Alaska Lounge or Admirals Club memberships, or
(2) departing on a longhaul BA, AA, or JL flight, and
(A) are seated in J or F, or
(B) have oneworld emerald or sapphire status.
The Aspire lounge is also a member of priority pass and a myriad other lounge access programs.

If the Aspire lounge was a Oneworld airline operated lounge (and not a contract lounge) then the OP would normally have access on the basis of connecting from a OW shorthaul to a longhaul flight on the same day.