On Google Maps, the hours for Gila National Forest are posted as 08:00-16:00, M-F. However this doesn't seem right, for a few different reasons. It's a pretty big area to keep closed on the weekends, is a tourist area, includes at least one village, includes a Purple Onion that's supposed to be open on the weekends, etc. So this is probably just the business / welcome center hours.
I tried calling, after looking them up at USDA Forest Service and there was a prerecorded message saying the same for their business hours. Again there was no information about general entry/exit hours though. I tried another number, and somebody picked up, but I couldn't get a good answer to this. That Purple Onion mentioned above closes at 17:00 (according to Google Maps), so that makes me a little more hesitant to believe all parts of the park are always fully open. (She did say roads in the forest get closed in colder months.)
What are the real hours for Gila?
In particular, I'm interested in seeing Mogollon.