Westward from the Trail of the Coeur d'Alenes, the route we took departed the trail at Harrison, ID, headed north on ID 97 on the east shore of Coeur d'Alene lake to where it intersected I-90, got on the freeway for a couple miles, and then got off to follow the Centennial Trail through Coeur d'Alene city into Spokane. From Spokane northward we used US 395 to connect to the Northern Tier/WA 20 at Colville. (Note: when we did this trip we were travelling west-east.)
Choosing the Colville-Spokane-Coeur d'Alene-Missoula route required us to make some compromises, like quite a bit of freeway riding in Montana. We found it tolerable, if not perfect riding. Thankfully the scenery was beautiful and the traffic not so bad. Others might not like it.