With help from sgordley's 2012 reports (thanks!), I tried out the South Branch above Flemington on an unseasonably warm November morning.
There are "No Parking" signs and fencing near the bridge/water treatment plant put-in he'd used, but I found another bridge just downstream near the YMCA's Camp Carr (take Hamden Road to Camp Buck Road) which worked off-season. Another option (described below under 'Directions'), is a third bridge 1/4 mile further down Hamden Rd.
From there it was an easy paddle down along a mostly debris-free, very scenic stretch of river, bordered by some rocky bluffs, wooded conservation lands, and a few horse farms. The river was broad with gentle turns, and alternated between 2-4 ft. depths and shallow, stone-filled riffles.
I was glad to have taken my plastic kayak (Dagger Edisto, 14 footer), as it often bumped over stones and rocks, but nothing serious, and only once did I have to push over them with my hands. Another foot of water would have helped, however (the water depth at the Stanton Station gauge was 2.5 ft.).
My planned take-out was just 3.6 miles down at the Stanton Station bridge (which has parking, but short path to the river would be tight squeeze for a longer boat), where I'd stashed a bike to taxi back up Hamden Rd. to my car. But as that only took 55 minutes, I continued downstream another 1.3 miles, past a train bridge and a short noisy stretch next to Rt. 31, almost to Packers' Island.
I was able to paddle most of the way back upstream to Stanton Station, but did have to get out and drag over a few shallow spots. I didn't see any good take-outs there (Packers' Island road is nearby, but would require crossing the train tracks), so probably the next one downstream would be Darts Mills.
Small parking lot (for conservation area) on the left by the bridge. Informal access at the bridge, down a four-foot open bank.
The put-in at the Hamden Rd. (third) bridge is under 20 minutes from Flemington. Take Rt. 31 North 5.4 miles, turn left onto Payne Rd. for 0.6 miles, turn right onto Lilac Drive for 0.8 miles, turn left onto River Rd. for 0.5 miles, turn left onto Hamden Rd. for about 0.6 miles.