While there isn't overnight parking, I'd highly recommend a hike to the Cascades Falls in Giles County. it's a little less than an hour from Radford. One link for this is: http://www.hikingupward.com/JNF/TheCascadesBarneysWall/
For an over nighter, perhaps Wind Rock. Don't have a link handy but it's near the Cascades and has a parking lot at the top of a mountain. it's right on the AT. Not super familiar with the area so not sure where any water sources are.
I'm pretty new at this.