Oak Flat Campground is a little off the beaten path. It's off of the Gila-Pinal Scenic Route, and it's a popular spot for OHV enthusiasts, but the OHV trails and roads are also great for mountain biking and hiking. There are a bunch of canyons in the area that can be fun to explore on foot, but if you decide to go for a hike, make sure to take a map with you since the trails and roads aren't marked super well.
Once you work your way south away from the campground, you start to get into the Devil's Canyon territory. The canyons back here are vast, primitive, and worth the trek to get back there.