If hot weather is forecast:

South to Lewiston

US 12 east over Lolo Pass along the Clearwater and Lochsa rivers to Lolo, MT.

US 93 south to MT 43

MT 43 east to Wisdom, MT. (Note: Wisdom enjoys just 45 frost-free days/yr. so midsummer's best. Also, the local VFW maintains a free, no-host campground w/a hut for restrooms & device charging and an open-air hand pump for a quick early shower. Also, also...mosquitoes and marsh loons.) 

Alternatively, ride south on MT 278 from Wisdom to Jackson, MT and get a cabin at Jackson Hot Springs Lodge.

MT 278 south to Dillon, MT

MT 41 north to Twin Bridges, MT

MT 287 south from Twin Bridges to U.S. 191

US 191 south to West Yellowstone for lunch.

US 20 west to St. Anthony, ID

If cooler (sub-90 deg.) is forecast:

From Spokane, cut over to US 95 

US 95 south along the Salmon River to New Meadows, ID

ID 55 south to US 26

US 26 east to I-84 in Boise

I-84 east to ID 21

ID 21 east to Idaho City (Note: About 3 miles before Idaho City there's a nice little USDA campground on the south side of ID 21 and a fantastic developed hot spring called "The Springs" a mile west of town.)

Continue east along the Payette River past Stanley to US 93

US 93 south to US 26

US 20/26 east to Idaho Falls

Then north on US 20 to St. Anthony

* Hell's Canyon National Recreation Area is a blast furnace in summer, even while riding along the Salmon, which is why I recommend routing based upon the heat forecast.

Ride tastefully!