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Pilot Mountain is not that far outside of Winston-Salem on 52. Nice views from the parking lot and a fun, if easy, trail around the Knob. If that doesn't get your fix, then head off to Hanging Rock. The views from Hanging Rock are great.
Stone Mountain and Doughton Park are also really scenic areas. These are west of I-77 above Elkin.
These were all normal day hike areas for me growing up in Winston. But from the Triangle, if I'm going to drive that far for mountains, I'll usually just drive a bit further to Blowing Rock. I love the section of the Blue Ridge Parkway from Blowing Rock to the Linn Cove Viaduct/Grandfather Mountain.
From Pittsboro, I'd guesstimate 2 hrs to Pilot Mountain and maybe a bit over 3 hrs to Blowing Rock. Either 87 to I-40, or 64->421->I-40. Hanging Rock or Stone Mountain would take somewhere between those two.