There are many trails in the Smokies that lead up to the infamous Mt. LeConte Lodge. The best one by far is the Alum Cave Trail. This trail reveals the most beautiful parts of the Smokies providing a variety of spectacular rock features and views, all-the-while offering an adventurous trail to the top. During rainy weather, the rocky trail offers the assistance of metal cables as things can get very slippery up there. There are also spots that are fairly narrow so those afraid of heights feel slightly more secure hanging on to the cables.
If you can get a reservation at the Lodge for an overnight trip, it's worth the strenuous trip 5 miles up to 6k in elevation. A day trip is also entirely possible as I have done but count on a little over 11 miles of hiking in one day if you decide to explore the gorgeous overlooks at the Cliff Tops and Myrtle Point.