Showtime7612: Welcome to TN! You are wise to inquire about the laws, rules, and regulations before you go out there! The old "Better to beg for forgiveness than ask for permission" doesn't fly far when you are dealing with LEO's.
The answer to your query is somewhat complicated. Where do you plan to hunt? I'm not asking for specific locations, of course - those are your business and not mine. However, there are different laws for different areas.
The Organ Mountains have private and public land. You will need permission from the owners of private land - same as everywhere else. Public lands are more complicated. And, when you're out in the field, it can be difficult to tell where one stops and the other starts - although that is important.
The Organ Mountains National Monument is off limits - period. The National Park Service law is simple: "DON'T."
There is probably some Bureau of Land Management (BLM) territory out there, as well.
Mining claims on public land are private as far as searching on them goes.