I lived in El Paso for awhile and did a lot of weekend trips in southern NM. Definitely a hidden gem as mentioned above. CO, AZ, and UT overshadow NM and that's what makes it great, IMO. I'll share some of my thoughts and experiences if you're looking for ideas.
I spent many a weekend in Gila NF. Forest Service roads abound. Lots of elk to see and secluded meadows for camping. The area feels like southern CO to me. This was all pre-2020, so who knows what the traffic looks like now, but it was uncommon to see another vehicle in the backroads of Gila. It's probably still the same - the place is so remote and you really have to go out of your way to get there. Thinking back to that time really makes me miss that place. Remote, quiet, and beautiful. There are also some really cool two tracks to explore around the Cloudcroft, NM area. One trail leads to an old abandoned solar observatory. The dome and telescope substructure was still there when I visited many years ago.
The Otero Mesa is special and desolate. South/SE NM. Big open grasslands with yucca forests and the occasional plateau and rocky outcrop. It's nicknamed the Serengetti of the Southwest due to its open expanses and wandering herds of pronghorn.
One of trips to the Otero Mesa, circa 2016 (I think).
One cool location to check out in that area is Alamo Mountain. Petroglyphs are everywhere. There is a spring at the base of the mountain and I suspect the mountain was a natural waypoint for natives crossing the mesa. The old Butterfield Overland stage coach route snakes through that area and one of its stops is at the base of the mountain at the spring. The rock foundation of the old rest stop is still there. Here's a link to a write-up I did over on ExPo about Alamo Mountain:
For ideas, this guy did a good job mapping some overland/Jeep trails in the southern NM area. I think I ran most if not all of his tracks. The NM BDR is also a fun route to explore, .gpx files for the BDR should be available online, but IIRC, it starts around Dell City.
Also check out Chris over at Venture 4wd. He's been living out of a pop-top Jeep JKU for years and has some good Youtube Vlogs from trips in NM: