The Shut-In Trail is a long, historic route which travels from NC 191 near Bent Creek and Asheville to the Mount Pisgah parking area. George W. Vanderbilt, builder of the famous Biltmore Estate near Asheville, constructed the Shut-In Trail around the year 1890. He used it to climb the Pisgah Ridge, linking his hunting lodge at Buck Springs below the summit of Mt. Pisgah to his famous Estate.
Although parts were obliterated with the construction of the Blue Ridge Parkway, parts of this trail still follow the original route. The trail was named by Vanderbilt for the ridge which it ascends, which was in turn named nearly a century earlier for the fact that it "shut in" the Bent Creek valley from the adjacent French Broad River valley.
Though the trail can be used as a long-distance route, it is most frequently used for short exercise and day hikes. This is generally necessary since overnight camping is not allowed along the Parkway (and 16.3 miles, with 3000 feet of vertical climbing, is a bit long for most day hikers). Those wishing to hike the trail with an overnight stop must plan a side trip onto adjacent Pisgah National Forest land to make this work.
There are many convenient access points where the trail touches and crosses the Blue Ridge Parkway, mainly at overlooks.
Not everyone limits themselves to day hikes, though. For over 20 years the Shut-in Ridge Run has been using the trail as an event venue, on the first Saturday in November. Joggers cover the entire 18 miles on this run. You can get more information on the official website.
The trail follows the same path used by the Mountains to Sea Trail for its entire length. Since you're close to the Parkway (never more than about 1/2 mile) and not far from Asheville, expect lots of hiker and jogger traffic on the trail.
Here is a map of the Shut-In Trail.