When I lived in El Paso, I explored a lot in the Cibola NF, Lincoln NF, and Gila NF areas. There is a lot to see in those parts and there’s not a lot of people, which is great if you like some solitude.
This guy did an excellent job mapping and recording some trails north of Las Cruces, NM.
Explore a wide variety of Jeep and ATV trails near Elephant Butte. All trails have maps, photos and GPS - gpx tracks.
www.jeeptheusa.com
The San Mateo GP loop is beautiful and, if you’re inclined to camp, there are a couple of establish primitive sites along that route, some even have camp shelters - as well as some excellent hikes. Lots of mining camp remnants and natural springs, really cool stuff. I used to fill up a water tank every time I would visit the spring in this area; it was some of the best tasting water I’ve ever found. Chris from Venture 4WD camped in this area about a year ago.
I’m familiar with some other cool hikes in Lincoln NF and the Cornudas mountains further south, but that may be a stretch from Sante Fe.