I'm not dissing Cherokee, I have wet a hook there many times, but you might also consider fishing other places nearby and saving the Permit fees. (Assuming you have a NC license). Lots of hatchery supported streams nearby like Scotts Creek, Alarka Creek, Deep Creek(stocked by NC for the last mile, changes to wild trout water in the park ) Tuckasegee River, which is floatable and has multiple species), great fishing for wild trout (and solitude if you are willing to walk for 15 minutes or take a boat ride) in the GSMNP streams
I rarely fish Cherokee anymore because of the crowds and the great fishing everywhere else. However, back in the day we had our best luck in the smaller, cooler streams this time of the year. The lower Oconaluftee starts to really heat up in the summer (I have caught smallies and catfish below Birdtown).
Bunches Creek is good, and Soco Creek upstream from the Casino is good.--Be mindful of hornets and yellow jackets if you fish any of these small streams this time of year. In addition to the flies Slickrock Tom mentioned, I like using a dry fly/dropper combo with a size 12 yellow foam stimulator on top and a size 14 to 18 dropper of general nymphs like pheasant tails, gold ribbed hairs ears, tellicos, and copper johns.tie my hares ears nymph using a tiny bit of peacock herl as the wing case and the fish love it. If I'm on a meat hunt, even a 1/80 oz crappie jig with a foam body and scented with a drop of anise oilhas resulted in a smoker full of fillets. I hope I don't incite a riot for mentioning the use of scents. You can use any lure/bait combo in tribal waters with the exception of the trophy section, and I would not personally use this technique anywhere but in a put and take fee area like Cherokee.
As for other places you might want to try , Straight Fork has good fishing, GSMNP regs apply, no tribal permit is needed, and is easy to get to if you are camping anywhere on Big Cove Road. Follow Big Cove Road to the Fork , go past the tribal hatchery and you are in the park. The road follows the stream for miles, you will see few other people and the stream is beautiful and has wild browns and rainbows mostly, but I have caught brook trout above the low water crossing (the road turns into a one way road descending from Heintooga Overlook on the Blue Ridge Parkway) If you fish upstream from this point, you will catch more brookies, and it is like being in the middle of nowhere. It's worth the trip just to be in this place, catching a few fish is an added bonus. I have done particularly well in the Summer here with these: a size 14-16 pheasant tail parachute, elk hair caddis, or yellow foam stimulator with no dropper needed most of the time.
The Oconaluftee upstream from the Visitor Center also has good fishing with no tribal permit needed, just watch out for the beavers..... I accidentally stepped on one while fishing here once. Scared the daylights out of me and him (or her?) both. I got soaking wet but at least I didn't get bit, lol.
A There are lots of other places besides the ones already mentioned, and you will be in the middle of some great fishing. Whether you stay in Enterprise Waters, the park, or any of the lakes or rivers, you have a lot of options. If you have a canoe or are willing to rent one from an outfitter and really want to have fun, do a float trip on the Tuckasegee River. It is considered transitional water below Dillsboro, so it is therefore marginal trout water, which is not necessarily a bad thing. I have caught trout, smallmouth, and big bream on the same float, and have seen but never caught some monster carp . I know others who have caught walleyes in this stretch, and there are big brown trout, flat head catfish, channel catfish and muskies here as well. There are over 20 fishable miles of water from Dillsboro to Fontana Lake. Anyway, hope this helps.