There are lots of trails in and around the Jocassee Gorges area. While there may be a locked gate or two, there are still miles of trails and gravel roads to follow.
Outside of the Horsepasture Road area, you may want to check out Whitewater falls, then head to the fish hatchery on Hwy 107. The kids should love both. Then, if you want some tame gravel road but great scenery, take Burrell's Ford Road to the Chattooga River. The river is where the Georgia Traverse starts. There are several short off-shoot roads off Burrell's Ford Road.
Good luck and have fun.