Just North of Deming, NM, there is a spot called Faywood Hot Springs. Open air hot mineral water soaking. Nice (reasonably priced) camping. "Clothes optional" camping and hot tub area for those so inclined. Take a short stroll to the top of "Observation Hill" for gorgeous sunrises and sunsets. Site used to be: a native American "healing site", Butterfield stage stop, mineral water bottling factory, hippie hangout in the 70-80s and now a mellow, out of the way, awesome camping spot with benefits.
Just North of Faywood Hot Springs is the Mogolon Rim/Gila National Forest and the Gila Cliff Dwelllings, well worth a stop. Silverton has some great history and was where Billy the Kid washed dishes for a living, fine Mexican food too!
While in Deming, find the "Adobe Deli" (somewhat of a challenge to find, good luck!) for an excellent steak dinner with a real "hunter cabin" atmosphere. Local favorite for a good reason! I have asked and been granted permission to sleep in their parking lot on occasion. Yes, they are that far out of the way....... Deming also has an interesting small history museum, which has a collection of artifacts from the Faywood Hot Springs.
South of Deming, is Columbus, New Mexico, where Pancho Villa invaded the US before WWI. Great museum, nice State Park for paid camping or camp for free at the airport. Excellent Mexican food in town! Just South of Columbus is the small Mexican town of Las Palomas which is fun to stroll around (at least it was 2 years ago).
All of the above are within 100 miles of each other.
Happy Trails!
Chuck in Arkansas