Steven has said most of it, especially the fact that unless we have a dry rest of the winter and an unusually warm spring, 4,000 feet is the maximum you'll get to in early June. Right now it's a dry winter, but that happened last year, too--all of a sudden in mid March, winter arrived and lasted well into June at the higher elevations. We won't really know until well into May what will be available. Watch Portlandhikers.org forum, especially the Trip Reports section, for what's going on. There will probably be a thread on snow levels, either in the General Forum or Trail Q&A, which we impatient hikers peruse carefully after each weekend to see how far up we can get the following week. The one item I'd take issue with Steven's list is that Jefferson Park is normally not melted out until early to mid-July, and then it's heaven for mosquitoes (the opposite for hikers!) for several weeks.
There is certainly plenty to do, though! The Gorge is always accessible (except in winter ice storms, definitely not a possibility in June). There are a number of intermediate-elevation hikes that normally become accessible in June. If you go to www.portlandhikers.org and click on the Field Guide (http://www.portlandhikersfieldguide.org/wiki/Main_Page), you'll be able to find a number of them. Click on the list of "Must See Hikes" and you'll find that all the ones up on Mt. Hood aren't accessible until July. However, if you click on "Mt. Hood Hikes," you will find quite a few that are accessible in June. They won't be up on the mountain, but they'll be nearby with views or other features. The wild rhododendrons bloom in June, as does the beargrass. From the Field Guide: Ramona Falls, Lost Lake and Butte, McIntyre Ridge to Wildcat Mountain (the Field Guide description shows the new trailhead from last summer, which printed guidebooks don't), Mirror Lake and possibly Tom, Dick and Harry Mountain should be available in June. Except for Ramona Falls, these are ridges close to Mt. Hood and feature views of Mt. Hood from various angles. In early June, if it has been a cool wet spring, there should still be wildflowers on Dog Mountain on the Washington side of the Columbia Gorge.
You might want to check out William Sullivan's "100 Hikes in Northwest Oregon and Southwest Washington." Some Mt. Hood-area hikes listed there that should be open in June that don't have PH Field Guide listings are Hunchback Mountain, Salmon Butte, Burnt Lake, West Zigzag Mountain.
In the June-accessible "Must Hikes" I would especially recommend Opal Creek, which will give you a close-up look at what Oregon lowland forest was like before most of it was lumbered off. Not only is the ancient forest awesome, but the deep blue-green pools of the creek and the clarity of the water are just astounding.
Bagby may be disappointing; the Forest Service has turned it over to a private company and they are tearing out all the picturesque old wooden tubs. They will, of course, charge admission. Aargh.