Lots of BLM and state land around Deming. You might want to pick up the New Mexico Gazetteer from DeLorme. Check with the local BLM office to see if there are any areas where camping is not allowed. I don't know who you would contact for the state lands but you can probably find out on the state govs website.
Same story on west to the Arizona line around Lordsburg. I think there might be a couple of places that overlook the Gila River.
If you get west of Silver City there are some roads that follow the Gila River back east that should have some sites.
There are some interesting rock art sites around Frying Pan Spring and Fr. Cummings.
The place to go in Hatch is Sparky's. It has a bunch of big caricature statues outside so its easy to find.
I need some wild