I believe the regulations changed and now local taxi firms are allowed to pick up from airport. Prior to this only cabs registered in Elche where allowed to pick up from Alicante airport.
I live in Torrevieja and Torretaxis.es are one of the companies where you can book on-line or ask for a quote. Minimum 48 hours advance booking. I'm sure there will be others in different regions who provide the same service.
I use Quickair as it works out cheaper for 2 people, the fare is now 6.92. The taxi fare to our house is another 14 euros.
I've yet to use Torretaxis but I believe the fare for us would be 50e for up to 4 people.