Early June in Montana

You could access the North and/or South Fork of the Sun River from Gibson Reservoir. Those valleys will be snow-free, though making a good loop is harder. The northern reaches of Yellowstone (in the park) would also be a good option.

I hadn't considered Yellowstone…we'll take a look at it. Thanks for the suggestions!

How soon do you think the Bob Marshell and Absaroka Beartooth areas would open up? Since I'm trying to work packrafting in I'm planning trips around water levels. I can hike in August but I can't paddle small creeks.

Quite a few loops in the bob could be done in early June, but it really depends on how warm may is. Also, rivers are often really high in early June. Scary high. The bear tooths will likely still be frozen.

The north side of the Beartooths (via Roscoe and Nye) opens before the southern reaches (off the Beartooth Highway) because the elevation is considerably lower. I went up to Mystic Lake in June a few years ago, although I think it was mid- to late-June. Depends on May weather, as well as winter snowpack. Three years ago, most of the Beartooths were inaccessible until late July/early August. The Beartooth Highway is always scheduled to open Memorial Day weekend, so theoretically you can access the AB from the road. However, I think people tend to do more skiing from the highway in June than backpacking.