Several years ago I looped through Idaho in late May/early June. It was a cool spring with a late melt out.
The popular state park at McCall had canceled a lot of Memorial day reservations. We came just after than and had no problems with a campsite. And the trilliums were out in full glory. McCall is a relatively high area.
BLM campgrounds along rivers were open. Stanley was still coming out its winter slumber.
We got snow flurries at Lost Trail pass (MT/ID border), and a bit later when we cross the Washington mountains east of the Cascades.
In sum, campground and highway access will depend a lot on altitude along with the weather.