Can vary wildly. In 2011, while riding a few days in the TA as part of a loop tour, we had nights between 35 and about 40 in Missoula, Darby and Wisdom, MT. The day we landed in MSO, over 1" of cold rain fell Darby, just missing MSO. That was June 29th. I also understand that the prevailing pattern up to that point had been abnormally wet and on the chilly side. Indeed, after the first day of riding we had clear skies and pleasant warm days. The nice thing is that even if the air is chilly, if the sun is out it warms you quickly.
Elevation does make a difference, In 2000, when I rode the TA from MSO to Fairplay, CO, there was morning frost at Coulter Bay campground in Yellowstone. The night before in Madison was also chilly. Again, that was during the last few days of June. But most of that stretch was pretty much how Josh describes it. Not much rain and nice temps. Definitely keep an eye out for thunderstorms, which can pop up seemingly out of nowhere. One afternoon I was napping in my tent in Dillon when hail larger than peas but smaller than marbles began to fall. Afternoon storms are one reason I like to start early.
Since I had started in the Pacific Northwest in late May I was carrying cold weather gear. I didn't bring anything like an extra sweater but rather slept in layers during the coldest nights.