Red Buttes Wilderness--might be a little early; check snowmelt progress first.
Possibly Kalmiopsis Wilderness (some of the trails wiped out by the 2002 Biscuit Fire are finally being reopened by volunteer efforts). The Illinois River Trail is open.
Farther east, Pueblo Mountains and Trout Creek Mountains should be open. There is also the Fremont Trail (contact the Fremont NF for info).
Do you have the William L. Sullivan guides for Southern Oregon, Eastern Oregon and the Oregon Coast/Coast Range? His Atlas of Oregon Wilderness is also excellent. These are invaluable sources for Oregon hiking, and they also include trips into northern California. While the guides are oriented towards day hiking, a lot of the dayhikes can be connected for longer loops, which Sullivan mentions in the text.
While the forum and Field Guide at portlandhikers.org are pretty much centered in the NW Oregon/SW Washington area, there are some reports from southern Oregon there, including a recent one on the Kalmiopsis. Actually, since Portland is 5 hours from Medford, even for me who has to walk the dog 20 minutes at every stop, the Columbia River Gorge should be included in a possible June backpack. Check the Field Guide on that site for the Mark O. Hatfield trail. You could do that end-to-end or just include it and Gorge Trail 400 (river level) as connectors for the many north-south trails in the Gorge.