re: Washington State in late June

Snoop Dawg's recommendations were great. Nothing there that will be snowed over.

Looking through my hiking logs over the past few years, here are some options of good hikes along with the probability that they will be hikeable...

Chain Lakes (Mt Baker) - 10%
Colchuck Lake (Leavenworth - Hwy 2 east side) - 90%
Franklin Falls (Snoqualmie) - 100%
Heather Lake (Mtn Loop Hwy) - 100%
Lake 22 (Mtn Loop Hwy) - 100%
Lake Serene (Hwy 2 west side) - 90%
Mt Pilchuck (Mtn Loop Hwy) - 75%
Naches Loop (Rainier) - 50%
Oyster Dome (Bellingham) - 100%
Ozette Triangle (Olympic Coast) - 100%
Park Butte (North Cascades) - 10%
Sauk Mountain (North Cascades) - 75%
Trappers Peak (North Cascades) - 25%

There's so many choices, so I can be more specific once you pick out an area you want to focus on. You could spend months here doing a different hike every day and not touch 5% of the trail options.

Anything less than 100% you'll have to check the trip reports to see what it's like. AllTrails and WTA.org will be the best sites for that. You can go back in history on those sites too for certain hikes and see the dates of when people said that the snow has melted.

I would highly recommend Artist Point if the road is open. You'll have to monitor that as well. There will still be too much snow up there to do much hiking, but if the road is open, you can drive see Picture Lake (last picture posted by AU) and you can see a few of those other sites, but you'll have to walk around in the snow to do it. It's possible it could be 70+ degrees and sunny. There very likely won't be hiking available, but you can walk around a bit and enjoy the views.

Mt Rainier area - Most areas still likely snowed over
Snoqualmie area - Hikes that don't climb a lot have a good shot at being available. High elevation ones will likely be snowed over
Mtn Loop Hwy - Most of these hikes should be available
Hwy 2 - Close to Mtn Loop Hwy. Same notes as Snoqualmie
Mt Baker area - Mostly snowed over. Decent chance of being able to drive to Artist Point and walk around, depending on if the road is open.
North Cascades - A few of the hikes should be available, but the high elevation ones like Cascade Pass, Heather Maple Pass Loop, Hidden Lake, etc will be snowed over.
Leavenworth - Should be able to do Colchuck which is a great hike, then grab dinner in Leavenworth
Olympics - Coast and lowlands will be 100% open. Higher elevation hikes will be hit or miss. I would think Hurricane Ridge area will be open.

The hiking crowds have gotten really bad Fri-Sun. Go during the week if you can, or go really early on the weekends.