Re: advice on which River to fish in June?
If you want to fish in the park, talk with LRO. You can't go wrong with them and they have excellent guides. I'm sure they can hook you up with a guide for tailwaters also.
If that doesn't work for you, contact Clinch River Outfitters. You can find them on the web I think, if not call a local fly shop in Knoxville called 'The Creel'. They assist with booking trips with CRO. I have floated with them a number of times and have never been disappointed. You will want to give serious consideration to floating one of the following: Clinch, Holston, South Holston. There are some other waters which are good as well.
If these options do not work for you, look at the bottom portion of the fishing report on LRO's site and it has local guide's recent reports. You may be able to find some help that way.
May you fish more than you wish for.