Sorry I am late to the party...........
HUMMMM 1st week of june... The area is not SUPPER busy yet so lodging should not be too much of an issue, but expect summer rates and no deals at any of the major hotels.
As mentioned the park is going to be slowish as the Sun road is open to Avalanche Creek, and though they have not in recent years but in the past opened up too the loop area by memorial day, though there is a bit more road maintenance being done preseason than in years past. . So you can get to Avalanche lake you will likely hit snow before you get to lake. Camas road is open so there should be some decent hiking out towards huckleberry Mt, and some wild life. If you get a decent rental (NOT little economy car) the road to Bowman lake is open though I have no idea how snowed in the tails up there are going to be, though I suspect even though we have had a bit of warmer weather snow shoes may be needed. Also there are plenty of roads free of snow out in the NF west of the North fork of the flathead river that you can go explore till you hit snow, then dismount an hike further up the fire roads. Also as mentioned many glacier is worth the drive and open but I would suspect many of the trails the start out of there are going to still be snowed in before you get to where you want to go. I have not explored much around two medicine, but again I would think there would be snow issues on the trials as they climb out of the valley. HWY 2 along the south end of the park is a fantastic drive and there are lots of places to pull off and gawk on both the park and NF sides of the road but again snow is likely going to be an issue in the Marias and Sky land areas, and likely around the essexs gate to the park as well, but I have not been home or up that way since early april and there was still a ton of snow
When I left for billings for work about a month ago, the West side of hungry horse reservoir was starting to clear of snow and you normally can get a good ways out that way by the 1st week of june, the 1st 12 or so miles are paved after you cross the damm, and from there on its rough, as is east reservoir road but IIRC the east side will get you deeper into the back country as the FS guys push the snow off that side early to get again lots of hiking opportunities as there are lots of fire roads that are gated or other wise blocked off to motorized travel but can be hiked for MILES. Down in the Swan river valley south east of big fork there should be lots of roads and trails snow free in the 6 mile creek and bear creek areas.
That should be enough to keep you busy for the month of june..... I would sugest focusing away from the park or the less travled areas of it like the NW area and west side like huckelberry Mt area out along the North Fork if you dont want to deal with throngs of people. Likely see few other tourists along the Hungry Horse, and even fewer down in the 6 mile creek area, other than a few locals looking for Moreals and a few ever increasing hippster millennials looking for the perfect instagram expierance to post and geo-tag to ruin for the rest of us, there should be much few people
