Will: I based my response on driving within a few hours of Albuquerque, since it takes about 3 hours on I-25 to get to Las Cruces.
There are some low-mileage backpacking areas in the immediate vicinity of Las Cruces like Aguirre Springs, or the long Sierra Vista trail (goes through some development though). Maybe Eugene or someone else has backpacked them to give you some idea of teh issues (I've only dayhiked or biked them).
With a couple more hours, there's the "bootheel" around Animas (the Mexican terminals of the CDT) or a little further the Peloncillos - border patrol will probably scout you out, but it's not a big problem. I went with 3-season gear and there was water in the Peloncillos (but that was 2007). Also the Southeast part of the Lincoln near Carlsbad for "Last Chance Canyon", "Rattlesnake Canyon", etc I've done with 3-season gear, .... plus the Guads. I've seen some pictures of backpacking after a snow south of I-10 (Deming to Lordsburg), so that could augment drinking water. Watch the property boundaries though.
Add that anything to the east controlled by White Sands and Ft. Bliss needs to be avoided as the military police have sensors all over the place.