Capitol Forest campgrounds, I ran across the Tenino City Park in USCAmpgrounds.info. A review of the Tenino town website (which doesn't seem to be current) confirmed the park existed but did provide any specifics. After our paddle on the Black River, we headed to Tenino, passing a large herd of alpaca along the way.
I didn't think to check, but I would expect to find good Verizon coverage in the area. The town looks like it could be interesting to wander around, and the park location makes the town easily walkable. Having said that, there are Walmarts in Chehalis and Tumwater if all you need is a spot to park overnight.