If the snow continues to melt as it has been the past few weeks, I think you will be fine to start in late June. I've driven both sections you are planning on. In ideal conditions they are doable with 2WD. Although, with the possibility of a rogue snow patch or two, I'd opt for an AWD or 4WD. Also, it would not be a bad idea to pack a chainsaw and shovel in the vehicle.
Here are some resources for you to check:
- The snow depth map here is not that detailed but gives an overview of whats going on. Click on the outlined Northern Rockies region on the map at the top. Then, click the Snow Depth map below.
- The Snotel sites give a good indication of specific location snow depth. The Banner Summit and Deadwood Summit snotels are on the route. So, check them. The Big Creek Summit and Secesh Summit ones are off route but will give addition insight into whats happening in the area.
Both of the above sites have historical data on them as well.
A couple of other things:
- The Middle Fork of the Boise Road is not that bad. Especially early season.
- Another option may be to ride McCall to Ketchum. Probably the nicest stretch of that length. Although not as many hot springs.
- That photo was taken at Burgdorf Hot Springs, north of McCall. It is on the Secesh Option on the Singletrack Options Map. There is a road that connects it with McCall. It's 95% paved. It's a great spot, and if you have a spare day, it'd be worth it.