My sister in law owned/managed a motel in West Glacier for close to a decade. They would have considerably more business after GTS opened - so August much busier than early June and not sure what predictions are this year.
When we left the park via W. Glacier in 2009 it was the first full day the road was open. The left turn lane from US 2 was stacked full of vehicles. That was late June. These days you need a reservation to enter the park in the GTS corridor during the peak season from around 6 am until early evening or so. That was implemented a year or two ago.