Not sure what you define as close, but the southern peninsula rivers are fairly convenient.  Thanks to Green Diamond Resource Co (aka Simpson Timber), access can be a bit problematic, especially on the upper middle sections of the Satsop and the Canyon river.  The Wynoochee gorge is a beautiful run, and makes a nice solo venture with bike shuttle.  The landslide drop is a bit more difficult than III+, but can be portaged RL.  And if you don’t consider the Wynoochee too distant, you might as well consider continuing over on FR 22 to the Humptulips, which has three really nice class III runs.  The upper narrow’s section is fantastic, narrow class III, with a fairly easy bike shuttle.  The middle “Falls” section is more class II / II+, with one class III ledge and what I consider a mandatory portage.  Again, easy bike or walking shuttle, but you’ll have to walk the 1+ miles up the dirt road from the take out, as it’s a gated road.  The lower Gorge run is class III as well, but you’d definitely want a second vehicle for the shuttle. 

You can also head north on Hwy 101 to the Hamma Hamma.  While close to (and between) roads, the class III section feels more remote than it really is, and it’s got nice variety.  Like the landslide drop on the Wynoochee, the short “horseshoe bend” canyon on this section of the HH probably rates more than III+, but you can scout and portage if needed.  Biking, jogging, or walking the shuttle for this section is easy.  Make sure to scout the take-out, as you won’t want to venture downstream in your kayak.  Walking downstream along the bank, however, is quite impressive.  There’s also the short section of the NF Skokomish below Staircase down to wherever the gradient tapers off in the lake.  It’s in Gary’s book, but I can’t comment as I’ve never run that section.