Earliest I've done it is late August.
I usually start trying to get into high areas in Paradise Park or Cairn Basin in early July but it's difficult to find the trail, and where the trail traverses a steep slope, there is icy snow at steep angle you have to walk across. I'm familiar with it and choose places I know are somewhat easier. Doing the whole loop would be difficult.
You might be able to hike around on lower elevations like Zigzag ridge. Or go up Paradise Park trail up to Paradise Park.
Probably Salmon-Huckleberry Wilderness on the other side of highway 26 would be doable. Or do something in Columbia Gorge like a Wahtum Lake loop. There are more trip reports at portlandhikers.org.