the early June season limits you…the Hoback will still be running off color most likely…the Upper Flat Creek is usually very good and a very fun creek to fish….String Lake(it is running water out of Jenny Lake) in GTNP is usually a good place early season.
I would say that it might be worth your while to bump over to the South Fork. I fished with WorldCast Anglers on the South Fork for four days this week, doing the guide school program. We actually caught fish on dries, streamers, and nymphs.