You may actually find a spot after 10 am!
Glacier National Park campgrounds fill up quickly -- but if you're coming to the park from the south east -- you might have a chance to get a site at the lovely Two Medicine. The entrance to the Two Medicine section of Glacier is separate from the main entrance, and it seems that many people miss this option until all the other options run out.
The campground is beautiful, and sits right on the lake with two mountains as a backdrop. Sites are spacious, however, there isn't a lot of privacy between sites. Some sites connect to others- which can be a bonus if you're traveling in a larger group.
We camped at Sites #72 & #73 -- two connecting sites one row back from the lake view. Each site had a fire pit, picnic table, tent pad, and access to toilets without a long walk. This site pair also had a nice trio of trees that we were able to use for our 3 hammocks (and hang them with a lake view).
To access other portions of the park like Going to the Sun road, the drive back out to the main entrance was an extra 10-12 miles. At the campground there is access to a number of short to medium hikes as well as a ranger station and lake tours by boat.