Ground Transportation tips from the Hyatt Concierge staff:  http://www.yattit.com/travel-tip/5972-Transportation-Tip-for-SA;  to share a cab is the least expensive.

Shuttle service to/from San Antonio Airport: http://www.saairportshuttle.com/  or call (210) 281-9900

Taxis are readily available at the airport, or see http://www.yellowcabsanantonio.com/?gclid=CNab5tStt5oCFSbxDAodyBv-bQ, or call 210-222-2222.