I've never geocached in the Las Cruces area, but I have geocached in the desert in AZ and in desert areas along I-40 in NM. My experience has been that the caches are not hidden a lot different than here, except there are few trees and those are pretty small. So hiding spots in rocks are favored. My recommendations for caching in the desert:
1) Carry plenty of water
2) Use plenty of sunscreen
3) Wear a hat - not a ball cap, but one with a full brim all the way around
4) Carry plenty of water
5) Watch out for rattlesnakes - especially in the evening and early morning before it gets hot - they like the rocks too
6) Watch out for chuckwallas - they will bite and not let go
7) Carry plenty of water
8 ) Be sure you have extra batteries for your GPSr and your cell phone is fully charged
9) Be sure your gas tank is full before you set out on your daily caching trip - gas stations may be few and far in between
10) Carry plenty of water
11) A vehicle with high clearance and preferably 4WD/AWD is highly recommend if you get off the pavement
While you're in that area, I recommend you head a little north and visit the White Sands Nat Monument. Neat place.
Oh, did I mention that you should carry PLENTY OF WATER? And don't just carry it - drink it!
"Wildness is a necessity." -- John Muir
"When you go to hide a geocache, think of the reason you are bringing people to that spot. If the only reason is for the geocache, then find a better spot." – briansnat