I live in Asheville and have been photographing wildflowers off and on for 30 years. I don't think that there is a "best time", primarily because of the elevation variations. In Asheville, elevation about 2300 ft, we get our early bloomers in mid March, whereas 30 miles (by car) at the top of Mt. Mitchell (elevation 6684 ft, highest in eastern US) it is still dead winter. The 2 most popular photography/tourist sites are the rhododendron "shows" at Roan Mountain (highway 143 at the TN state line) and Craggy Gardens (Blue Ridge Parkway near Mt. Mitchell) which usually peak the second or third week in June. Any time between those dates will depend on the elevation and weather (which can be quite variable). My suggestion would to be to visit western North Carolina anytime from April to late October and you will find plenty to photograph.