Start by visiting this site for NC dept of natural resources,
Trout Fishing (ncwildlife.org). NC has a very developed and multi layered Trout program. Use the maps on that site to see where they are managing trout waters (stocking) and learn the rules for the waters in the area you are fishing. Make sure you have a NC fishing license with a trout stamp. Then use the maps and guides on the site to locate your target water, and understand the rules for that water.
If you don't mind fly fishing the best waters will be designated delayed harvest and are fly fishing only, waters like the Davidson river, White Oak (tributary of the Nantahala) and many of the other delayed harvest waters get a very large % of the stockings. The maps will show you. You can even see how many trout, which variety, where and what day they are released.
Side note, June-Sept trout fishing is very very hard even in the mountains due to the higher water temperatures.. The delayed harvest waters are mostly fished out (intentionally allowed before water temps rise to dangerous levels for trout) in those months and new stockings do not begin again until Oct. You will have to find high mountain creeks/streams for native Appalachian brook trout (small but pretty) or fish deep in the Lakes. You can find some deep holes in the rivers where the stocked trout that survive the heat are sheltering, but they aren't biting very often. Late Fall and Early Spring are prime trout months in the mountains, the water temp is cold enuf for the trout to be healthy and the state of NC stocks during those months.
Good luck. I have fished the mountains of NC for trout over 20 years now, its my favorite past time. I have traveled a bunch to fish trout and salmon on the fly, from all of the Rocky Mountain states, into British Columbia, Alaska and also Maine, where they have the largest and healthiest native, non stocked Brook trout in the lower 48. The mountains of NC/tenn are still one of my fav spots to fish.