Your best bet is putting in at Bradford Fishing Access on the Pawcatuck River (Westerly), and paddling up to the Burlingame Canoe Camping area. It's about a 3.5 mile paddle upstream, all flatwater.

Another option is Stony Point on Worden Pond in South Kingstown, either paddling down the Chipuxet River from the Taylor's Landing access to the pond (although you'll have some beaver dams to climb over in both directions) or putting in right in Worden Pond off Worden Pond Rd. I don't recommend it now, though...there is a toxic blue-green algae advisory for Worden Pond now.