Chiricahua is one of my most favorite national parks/monuments/ect. I ended up there by accident on a trip. One of the best parts of the trip.
If you haven't already hiked the The Big Loop do it. It is about a 10 mile hike so a most of a day activity and is a long up hill at the end regardless of the direction you go. So not a easy hike but it hits all the best parts of Chiricahua. The balance rocks are amazing.
That is the only place I have been that didn't feel like earth. It really felt like an alien world
Also if you have a vehicle capable enough. Take the dirt road over the mountain to ~~Page~~ portal. There is a little bar/restaurant there that isn't bad. And the drive is great. Assuming there hasn't been any serious changes to the road. Anything Subaru Crosstrek or better should be fine.