A great hike to do while you're up visiting Clingmans Dome is this 3.6-mile (round trip) hike that starts near the visitor center. It's mostly downhill on the way there, making for an uphill slog on the return. When you reach the bald you'll be treated to some of the most spectacular views in the park.
Go early in the day (before 9am) to avoid the crowds. Although you can hike it any time of year, the road to Clingmans Dome closes in winter (December to March). Bring crampons if you come then.