Doesn't look like runoff is going to be as lengthy or severe as last year so basing out of Gardiner may be a viable option. Not sure if the Big Three in the NE end of the park will be fishable, SC is usually first to clear so that may work; streams in/around town may be fishable as well. Over on the west side, Firehole's usually thermally challenged by then and probably not worth fishing (fish would be too stressed anyway), Madison (in the Park) and Gibbon could still fish well, Madison outside the Park should be fishing very well, and Henry's Fork is not that far away from West Yellowstone if you want to give it a shot. If you are on the west side I'd cast a vote for Blue Ribbon Flyshop if you're looking for a guide - great folks there.
Regards,
Scott