Snow could be a problem in June. It really depends on the winter. If I recall ice out on the lakes is early June, or at least was back in the 70s. Of course that was 30+ years ago.
You could ping NOLS in Lander and see what they think of early June access. Also call the ranger station in the area before your trip. They have probably done a fly over of the area and can give you an idea of snow pack (if any).
Are you set on the Winds or would you consider other areas/states?
The reason I ask is that Kebler Pass near Crested Butte in Colorado is supposed to rival it's fall beauty when the aspen leaves first come out in the spring. I haven't experienced spring there personally only fall:
The cool thing about Kebler is that it has the largest aspen stand in the 48 states. Big... real big. The trees are really old too. Some of the tree graffiti I have seen dates back to the 20s. And an 80 year old aspen is pretty tall. They have large ferns in the some of the groves as well. Pretty cool area and one of my favorites in Colorado.
Anyway, I will be in Lander in the next week or so and can at least tell you what the snow pack is around town.
BTW, if you do fish and happen to visit the Winds in June there is usually a free fishing day. Last year it was June 2nd (http://tinyurl.com/3fphv5). The Winds has great fishing, hence the question :)