Roaring River Wilderness and Bull of the Woods Wilderness would be the best local bets. June should see both of those areas start to melt out, but its right on the edge. And yes, of course bugs will be bad.
Both of these areas have a network of trails for plan-as-you-go hiking. They both have several lakes you can poke around and explore. And neither location is especially popular (relatively speaking).
You should also consider Mt St Helens as an option. There are a gazillion trails in and around MSH, many open around May/June, depending on elevation. See here for starters, but you really need a set of maps to see all of the many possibilities the region has:
https://www.wta.org/go-outside/seasonal ... lens-hikes
Areas just south and east of the mountain itself are usually accessible early. If the entire area is open, you can do the Loowit trail around the mountain and/or Mt Margaret areas to the north.
Nearby Green Mountain has a scenic 3-day loop with possible extensions.
If you want something further out or earlier access, Rogue River and Umpqua River would make good choices.
Trip recommendations, current conditions, and other trail related Q&A