I have updated the photo guide for June for the news, access and conditions in Mt. Rainier NP, along with some updates for seasonal information for other pages. More updates for June are forthcoming within the next week on some of the topic Web pages about the seasonal conditions.
June is the month everything changes in the NP, both as the snowmelt continues through the month, slowly moving the snow-free elevation from the lower to the mid and upper elevations. The lower than normal snowpack was later peak snowpack and onset of snowmelt and is close to the normal time trend and should be completely snow-free in the 5-6,000 foot elevation sometime in about the second week in July (my forecast).
June, from late May, is when the roads clear and the facilities open for the summer visitors season, from Memorial Day through the Labor Day weekend. By mid-late June everything will be open except the Sunrise Road and visitors center and the Mowich Lake road, currently shut at the Paul Peak trailhead. These open in early July before the July 4th holiday weekend.
That aside, even with the best opportunities for hiking and photography, it's generally still cool and frequently rainy, so as I always mention, be prepared for all weather conditions and be flexible with your plans if the weather turns bad. This is standard practice for experienced hikers and backpackers, but this is important even with day hikes because you could be caught miles from the trailheads in a rainstorm.
That said, go and enjoy the NP.