I have ridden the entirety of GTS west to east once and up and back down the west side two other times. The W-E crossing and the second up and back down were both on the first day the road opened to Logan Pass.

You shouldn't have any problem assuming there is no late storm when you are prepared to cross. They do a great job of clearing the road completely. The only thing we encountered was nowhere to pull off at Logan Pass the first time. There was so much snow that the parking lot was not cleared and the visitor center was still burried nearly up to the roof. Somewhere I have a photo of me leaning against a 6'+ high snow bank at the summit. The real issue will likely be water on parts of the west side. Melting snow runs down the hills and can make the west bound lanes wet in places. Have you ever seen photos of the "Weeping Wall?" It's a section where the water poors down a rock face. I will try to post one later tonight.

If you plan to cross in June, the issue will be one of timing. Back in '09 the GF and I came from Waterton Village over Chief Mountain to St. Mary and intended to cross around June 24 or so. The entirety of the road was not open. While it had been plowed, a major avalanche over the winter required significant repairs. A year or two later, there was so much snow that the pass did open until some time in July. It may have been as late as mid-July.

You should also be aware that there is a multi-year road re-hab project going on. In order to have some uninterrupted construction time, the park has been setting an earliest opening date for crossing Logan Pass. Even if the road is cleared and ready to be driven in its entirety, it will not open in its entirety until the earliest date. You should check the official park web site later next year for details in case they plan to do that again. You can also follow plowing progress once it starts. If you have never seen the photos and videos they post, they are pretty cool. You really get a sense of enormity of the effort required and the dangers involved. And sometimes they even encounter critters during their work. Here are the photos from 2012:

http://www.flickr.com/photos/glacier...7629725011599/

If you get stuck in St. Mary you can wait it out or go around the park via U.S. 89, Looking Glass Hill (MT 49) and Marias Pass (U.S. 2). That's what we did in '09. Made it all the way to Sprague Creek Campground inside the park. 100 miles. But it was the only way we would have had time to ride the west side and still make our flight out of Whitefish a few days later.