We had a ranch for sale in Western Val Verde County off Dolan Creek road that had a lot of elk on it. There is a suprising number of Free range elk all the way from Ft. Davis east thru to Edwards/Val Verde County.
Word of Warning if you do shoot one: Elk are big animals and retain body heat very well. If you do get one during warmer weather, get to it quick and open it up immediately and get the hide off ASAP.
I saw a entire bull elk spoil after a bowhunter shot it in September and waited an hour to go find it after he shot. Imagine seeing 1/2 a elk carcass that was lime green.