Bear Creek Trail, out toward Ellijay, is legal for bikes and the most ridden singletrack on this list. The Pinhoti Trail sections, Mountaintown Creek Trail, and Windy Gap Trail see regular bike traffic too, but the wilderness boundary runs close to some of those routes, so check the specific trail page before riding. The Forest Service roads climbing Potato Patch Mountain, on the Holly Creek side, are also a local favorite and carry no wilderness restriction at all.
Mulberry Gap, an adventure basecamp about twelve miles from Ellijay, is the closest thing this area has to a mountain biking hub. They run shuttles, rent cabins and campsites, and sit at the center of what Georgia's tourism marketing calls the Mountain Bike Capital of the state, with direct gate access to Pinhoti, Bear Creek, Mountaintown Creek, Windy Gap, and a long list of other trails and forest roads. If you want current trail conditions or a shuttle instead of an out and back, they are the people who know.