There are as many day hikes in the Smokies [1] as there are days. Below are some of the most highly recommended choices.
Links:
[1] http://www.moon.com/destinations/tennessee/great-smoky-mountains
[2] http://www.moon.com/destinations/tennessee/great-smoky-mountains/recreation/hiking/day-hikes/loop-hikes
[3] http://www.moon.com/destinations/tennessee/great-smoky-mountains/recreation/hiking/day-hikes/waterfall-hikes
[4] http://www.moon.com/destinations/tennessee/great-smoky-mountains/recreation/hiking/day-hikes/hikes-view
[5] http://www.moon.com/destinations/tennessee/great-smoky-mountains/recreation/hiking/day-hikes/casual-strolls