Any of the river valleys in ONP are likely to be a good bet….until you start gaining elevation.
I’ve done:
Early June — a loop from the North Fork Trailhead (west of the Quinault River). There was snow from when we climbed out of Elip Creek to down around Three Prune.
Late June — Hoh River Valley. Snow-free, but we only went as far as Elk Lake (there were a lot of climbers going further up that way to summit Olympus)
4th of July — Enchanted Valley, snow free to the Valley (e.g. where the old Chalet stands), but I met folks who wanted to go further who had to turn back because they weren’t equipped for snow.
So, nothing is quite a sure bet. I’m in Portland. Driving distance might be the variable that has to give to make this work. If I were planning a weeklong trip in the mountains for June, I might research options in the Trinity Alps of Northern California.
You might also check out the Blue Mountains of Eastern Washington/Oregon…never been, and someone else here (Ian) didn’t have the best experience on the trip he planned there last spring (don’t do exactly what he did)…but there may be some viable options there too. It’s on my “early season” list.