Getting to Abrams is the bigger problem. It seems you can never allow enough time to get from the picnic area to the trailhead parking lot because of all the sightseers.
I second the Elkmont recommendation. It's quite an experience. For something a bit more physically demanding, try the West Prong of the Little Pigeon just above Sugarlands, or anything up to the Chimneys trailhead and Road Prong... great visuals, limited hiking and fantastic fishing at the right time of the year.