There are two possibilities, but neither is very appealing or easy to see.  You'd need to start at the top and work your way down in order to find the takeout so that you'd recognize it from the river.  I've taken out on river left just up stream of the bridge.  There is an overgrown, steep, muddy trail that winds back up to the highway.  It comes out in some of the new landscaping they did when they "repaired" the bridge last year.  If you hop the guardrail on the East side of the highway south of the river and follow your nose, you should be able to find the trail.  Again, it's harder to find from the river.

THere is a stream that pours out over a sandstone waterslide like waterfall on the south west side of bridge (river left downriver side) I have been eyeballing this for a possible put in, but the landing is a bit sketchy....and maybe a 50 foot drop...

Not exactly on subject but I found this funny. I was showing a friend that I am taking down the lower green today where Flaming Geyser is on Google Maps. As we zoomed in guess what I saw in the river.... A raft from satellite !!! pretty impressive, looks like an NRS.

http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=green+river+wa&sll=47.472663,-121.816406&sspn=11.736977,18.61084&ie=UTF8&hq=&hnear=Green+River,+King,+Washington&ll=47.276256,-122.01898&spn=0.001438,0.002272&t=h&z=19

Scroll a little further upstream, and there's a yellow raft, too.  Where are all the kayakers?    I bet you they'd show up with a little better resolution.

That used to be my favorite fishing spot before I started kayaking!

I'm pretty sure when they redid the bridge they also made a road all the way down to the river.

It has easy access to the river and a great swimming hole.  I know it's more popular now... I couldn't keep it secret for 11 years... darnit!

I'm sure you could carry down there easily, we've tubed down to the Yo-Yo stretch from there.