Update for anyone following the action:
Hitched yesterday after getting off the trail. Had a small emergency back home (now resolved) that forced us to accelerate the last leg of our journey; we originally planned to hitch this morning and drive back this afternoon, which would have been much easier, most likely.
Started at 4:20pm at the bike trail west of Apgar. "Polebridge" sign in hand, we got lots of waves from folks who seemed to be saying they weren't headed that far. We were picked up around 4:50pm after 100 or so vehicles by a younger couple heading to dinner in Polebridge. They dropped us off at the Mercantile and we started walking the 2 miles to the Polebridge Ranger Station. Didn't even make it 200 yards before the first car to drive by picked us up (another young couple) and dropped us off at the Station. They were headed to Bowman.
The Kintla and Bowman Campgrounds both still had vacancy, so we thought we'd try to hitch until Kintla filled or 20 minutes, which ever came first. 20 minutes passed and no one was headed to Kintla (3 or 4 vehicles passed by heading north). We had stashed a mountain bike at the ranger station for just this reason and so I grabbed it and started the 15 mile ride north. My wife stayed at the inside road fork and continued to stick out her thumb for another 45 minutes, thinking if she caught me we'd just exchange the key and I'd wait for her to drive down. She said about 15 of 20 vehicles stopped during that time (5:50-6:30pm), all were heading to Bowman and nearly all seemed willing to pick up a hiker without a second thought. None headed to Kintla, sadly and the Kinlta campground filled less than a half hour after I left.
I ended up biking all the way to the parking lot at Kintla in about 70 minutes, and no cars passed me. Felt pretty good about it too, especially after 12 miles of backpacking that morning/early afternoon. Zipped back down, picked up the wifey and made it to the Mercantile with 10 minutes to spare and grabbed a stick bun and a cinnamon roll. [Sorry, no photos.] They were closing early (8pm) for a square dance, which we didn't stick around for since we were starving by that point.
In summary, we did very well considering our unbelievably late start from Apgar and the wisely stashed bike. I think hitching on the west side is a viable option, especially with an early start. Most would probably find it a 2-5 hours hitch if they started around 8am from Apgar. We spoke to a couple groups on the trail who hitched from Apgar to Kintla or Bowman successfully, all using the outside road - some in 2 or 3 rides to Polebridge, others in only 1. A ranger we spoke with at Polebridge thought hitching prior to 2pm to Kintla would be successful on most days, but after that is a crapshoot. I agree. Bowman would be a much easier ride to catch, obviously. The opposite route (to Apgar) would be as easy if not easier in my opinion. A much higher proportion of folks seemed to stop on the dirt roads in the park. Getting picked up at Apgar seems to be the crux.