The S Fork of the American has some campsites where you could stretch it into an overnighter. We've always ran it as two separate day runs, but there's no reason you have to. Chili Bar run on day 1 and the Lower Gorge on day 2.
If you don't mind flying into Albuquerque or Santa Fe, then I'd recommend the Chama Canyon. Class II/III- with some excellent scenery and camping. Water flows are dam releases from El Vado, so there is often boatable flows on weekends well into the summer (the dam is opened up generally from about noon on Fri through noon on Sun, but if you're far enough downstream, you can boat it well into the afternoon on Sun). I'm even run it mid-Sept before. Its about a 2-1/2 hr drive from Albuquerque to the Big Eddy take-out.