CS -
You don't offer much info on what you want to do or how you tour.
Are you willing to do some dirt or do you need all pavement?
(That is more applicable in the west than the east - but it may impact route choice.)
Also, will you be camping or do you need motels every night or a mix?
(Again, in the west it can be tricky getting motels in remote locations.)
Also, how direct of a route do you want? How much time do you have and what is your daily mileage?
Are you thinking about wandering via Yellowstone or do you want a straight shot?
Thinking backwards from Missoula:
The ACA TransAm Route links Lost Trail Pass on the Idaho border to Missoula.
If you look at an Idaho map you will see three diagonal roads south of Lost Trail Pass.
Hwy 28, County roads from May to Howe, & US 93. All of them are fairly remote.
US 93 is shortest, but has no services for miles and is narrow with moderate traffic in places.
The country roads include dirt stretches and are extremely remote - only for experienced tourers.
Hwy 28 has low traffic and country stores at crossroad hamlets. And is very scenic.
But there is a "But" - Why is there always a "But"?
Connecting from the south is way easier via US 93 than Hwy 28.
The US 93 connector is via US 26 from Arco to Blackfoot.
Taber Road, thru the almost ghost town of Atomic City, is almost carless with varying pavement.
The Hwy 28 connector is via county roads from Hwy 33 to Idaho Falls, US 91.
This entails busier roads with a good deal of local traffic.
Yet another possibility is to combine the best of both - -
Hwy 28 from Salmon to Hwy 22 then south on Hwy 22 thru Howe to US 26.
A bad fire a few years ago destroyed what was left of local businesses in Howe.
From Blackfoot you might take US 91 and Hi-Line Road to Pocatello -
Then follow Old Hwy 91 to Virginia and Westside Road to Dayton and Weston.
(Remember - - I am going backwards from Missoula south.)
This gets you to the Utah border near Logan.
Here is the link to Idaho DOT traffic counts:
https://itd.idaho.gov/highways/roadwaydata/Maps/ATR_WIMmap_map.html(Click on the traffic count location for multi-year data.)
Under 1000 - good; Under 500 - Super
1000-2000 - okay; Over 2000 - Tricky without a shoulder
Over 4000 - Dangerous without shoulder