You could combine some other trip(s) in Olympics (e.g., Ozette loop) with a trip in Enchanted Valley, southwest corner of ONP near Lake Quinault. Sounds like what you’re looking for: rainforest, huge trees, up along a river valley with little waterfalls everywhere, 13 miles in to area they call “Little Yosemite” b/c of steep cliffs and waterfalls. Plus early July is great time to go.
Use wta.org as resource, e.g., here’s the dayhike listing for Enchanted Valley trail, which would only go a few miles in. There are loops you could do past camp at 13 miles in, but there might still be snow in the passes.
http://www.wta.org/go-hiking/hikes/quinault-river
Best resource for multi-day hikes in WA is Craig Romano’s “Backpacking Washington” book.
Here’s an Enchanted Valley trip report, you can probably find many others at nwhikers.net:
http://www.portlandhikers.org/forum/viewtopic.php?f=8&t=12152
There is a high concentration of bears in the valley, at least in spring/early season. But bears in Olympics are not problematic, bear hangs at most main camps, otherwise rangers will tell you you have to have bear can (which you can rent from them for a few bucks).
Another option in Olympics would be the Press Expedition Traverse. A mostly lowland hike (45 miles) that starts near Lake Quinault/southwest ONP and follows Elwha River from its source to where it empties into Strait of San Juan de Fuca. Need shuttle since it’s a point to point hike. Here’s trip report with some good forest pictures showing southern leg for this trip down near Lake Quinault:
http://www.nwhikers.net/forums/viewtopic.php?t=8004821&highlight=press+expedition
Here’re pics from same person of January hike of the northern section of Press Expedition Traverse:
http://www.nwhikers.net/forums/viewtopic.php?t=7996229&highlight=