I was on the Trans Am heading east for a few days last year. Left from Missoula on June 29th and rode to the eastern base of Big Hole Pass. On July 4th we spent the night in Twin Bridges, which is also on the Trans Am. We met several western-bound people who had started early to mid-May.
As noted, conditions can vary from year to year. When we arrived in MSO is was rather chilly and the weather had been not so nice all spring and summer up until then. The week before our trip, snow showers were forecast in some of the areas we were to ride to. Fortunately, we hit the first extended stretch of dry, warm weather.
Having done the the entire Nothern Tier, the western section to Glacier a second time and much of the OR section of the Trans Am to Florence, I can recommend taking the Trans Am to MSO and then jumping on the Great Parks to Whitefish. Form there, head to Glacier, make camp and ride up and back down the west side of Going to the Sun. Then head back west and finish on the Northern Tier. Sort of the best of both worlds. You would miss McKenzie Pass in OR, but GTS more than makes up for it IMO. Plus, I wasn't a big fan of the high desert of eastern and central OR. The WA mountains were more interesting to me. From near the end of the Northern Tier you can pretty easily get to Seattle, which offers plenty of transportation options home.