Yeah there really isn't anything to do around here. Boring as all hell!!
Since you are going to Alamogordo, as was previously stated, you can check out White Sands. Ruidoso is a nice area in the mountains, and you can walk the shops or go hiking. Don't really know what there is to do there. T or C has Elephant Butte, and don't know to much about other things there either. I'm originally from Artesia, so I don't really go outside of straight there and back to LC when I'm driving. Definitely boring as hell, but you might find some awesome things to do. Sorry you have to come out this way though.
Oh, and depending on when you come in, some of the guys from the EP/LC Chapter get together in El Paso on Wednesday nights to hit the track I believe. So if you are here!!