I spent two days in the Smoky Mountains, a Monday and Tuesday in April, and was fortunate enough to avoid the notorious crowds. On the first day, I did the Cades Coves loop. This takes you through a secluded valley deep in the park, with bright green pastures and historical wooden buildings. It's also known for its wildlife, but I didn't encounter more than wild turkeys (huge!) and deer.
About halfway on the loop lies the start of a popular walking tour, the Abrams Falls Trail. It's a 5-mile walk. It takes you on a sometimes slippery path through the forest, ending at a waterfall. Not wildly spectacular, but a fine way to stretch your legs. I walked it in about 3 hours in total.
The second morning I took the Newfound Gap Road that leads right through the National Park, from Gatlinburg (Tennessee) to Cherokee (North Carolina). Along the road are a number of overlooks that give good views over the forests and the mountains. The latter indeed with the layers of "smoke" attached from which it takes its name.
In the evening I undertook the last attempt at spotting a bear (or other wildlife). I went back to Cades Cove, where I arrived at 7.15 pm. The setting sun gave the open fields a wonderful golden glare. There were more cars around than during the day, many of those belonging to photographers. But again no bears! Only deer.
The hosts of my B&B told me when I came back that (young) bears when they are hungry even come into Gatlinburg and spook around the house. At this time of the year, they are not often seen. Cades Cove at dusk however still is the best bet.