From GSO, I will second Matt with Stone Mountain. It is 1.5 hrs from Greensboro, but very much worth it.
Pilot Mountain has some nice trails. You can drive to the top and hike around, or park at the bottom off Pinnacle Hotel Rd and hike all the way up. It is a fairly long, fairly strenuous climb.
Not a ton of hiking trails there, but Fall Creek Waterfall up off Deshazo Rd is nice. You can hike past the waterfall and walk along the Mayo River for a little distance.
There are a lot of hiking trails around the lakes North of Greensboro too that are very nice and obviously close by.
Rock Castle Gorge hiking area off the Blue Ridge Parkway is very nice, but probably too far for your timeframe.
Chinqua-Penn Walking Trail is nice and not far, but not a long trail. There is a cool old dam there. I've only been there once.
Cumberland Knob Gully Creek trail is probalby the closest hiking trail on the Blue Ridge Parkway to GSO at about 1.5 hrs.
I've only hiked at Uwharrie once, Many years ago, and I don't remember what trail I hiked.