Well, last time I was in Glacier NP a few years ago, in June coincidentally, Going to the Sun Road was closed for a day because of snowfall. So, make sure you're ready for any kind of weather because it can snow on you at any time.
And the storms that time of year can roll in with some wicked wind and rainfall/snow, so be prepared. Also, I'd say have a nice backup plan or two for possibly a lower elevation hike if the road is closed or certain trails are still inaccessible.
Temperatures can vary too. When I was there last, it varied from freezing or near freezing temps when it was overcast(not including wind chill) and nice and comfortable when the sun was out and the skies were gorgeous.
Obviously there are the precautions about bears. Simply said, when you get there, pay attention to and heed the advice the video tape you'll watch gives you and keep your eyes, ears and nose open all of the time. Chances are, the bears will be looking for left over winter deaths and eating a ton of roots and berries. Pack some bear spray and enjoy. Don't let it ruin your trip, just be aware of the dangers.
Too, I'd recommend and few days before your trip, you call the the park's backcountry office and see how conditions are concerning weather and trails. You don't want to watse all of that time going out there and finding you can't do anything. I almost had that happen to me this last September when I was trying to go back to Denali NP one more time before it snowed. The area I'd planned on going, the same one I'd been in two weeks prior, had just gotten six inches of snow and the buses no longer went back that far. Had I not called, I'd have gone all the way down there only to be turned around.
My advice is pretty much this. Be prepared for just about anything because just about anything can still happen that time of year.
Whatever happens though have fun and enjoy the scenery because there is little else that compares to Glacier.
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In all things of nature there is something of the marvelous.-Aristotle