It's been really dry and hot, check the TFR's on your flight app as there are some significant fires raging in the mountains surrounding the plains right now. Check
https://inciweb.nwcg.gov. Mostly in the Gila and San Mateo Mtns. I've done some passage through the fire areas and the smoke does make it instrument flight even if you're not in the TFR. But you can get to the other side.
The Very Large Array radio telescopes span the plain is of interest. Never landed in the plains but like you am curious to explore the area.
Check with
https://www.nmpilots.org. There are numerous Forest service airstrips available as the group is building an aviation tourism program similar to Idaho et al. Poke around on their website and you'll find a listing of airstrips that may suit your plans and interest. We've been busy flying mostly to our place in the Zuni's so I don't take opportunity enough to enjoy other destinations as much as I would like. I have been to Negrito in the Gila's (a great camping location with toilet facilities and crosswind runway) but there's a fire there now. It would be a great place. I think the NMPA website would be very helpful for your plans.