My limited experience with national parks is that the campground host will likely have a direct phone line - however, they will never give that out.
If there is a visitor center at that location - they will also not give out that phone number.
The staffing simply isn't up to the level needed to answer constant phone calls.
Glacier, I'm told by rangers who work there, uses reports from the volunteer camp host every hour or every two hours to report the # of sites occupied and # of sites open each day.
This allows the main information number to have pretty accurate on site availability at the time they receive a phone call.
Of course, they don't know how many people are already on the road to the campground. One time I called in to report our campground at Big Bend NP was full - and found there were seven groups of campers already on the highway up into the mountains headed to our CG.
Glacier's historical information on the calendar at the bottom of this page -
Two Medicine CG - is a very good feature. You can check each month last year, or the last couple years, and get a good feel of when you need to be at the CG.
Though of course it gives you no information about the availability of the larger sites.
Full-Time 2014 - ????
โNot all who wander are lost.โ
"You were supposed to turn back at the last street."
2012 Ram 2500 Mega Cab
2014 Flagstaff 832IKBS TT