An air carrier Captain related the debilitating effects of fatigue resulting from back side of the clock scheduling including extended breaks between flight segments.

Narrative

I was assigned an all-night trip originating in ZZZ1 to ZZZ2 with a report at 1900. [We] arrived in ZZZ2 at 2111 with a 02 hour and 35 minute ground time before continuing to operate the 'red-eye' from ZZZ2 to ZZZ3. I afforded all opportunity to get plenty of rest to prepare for the trip. I stayed overnight at a nearby hotel; checked out at 1730 and proceeded to the airport for a 1900 check-in. I completed at least 12 hours of total sleep prior to reporting for the trip. Upon arrival at ZZZ2 from ZZZ1; I felt pretty good during the 2 hour and 35 minute ground time. We left ZZZ2 for ZZZ3 at 2345 and both my First Officer and I felt alert. About half way to ZZZ3; both my First Officer and I experienced the onset of fatigue.Symptoms included yawning; short naps; head nod; and reduced concentration. We missed several radio calls from ATC. I attempted to combat fatigue by placing the overhead cockpit lights to bright; but this reduced our outside visibility; then I selected DIM on the lights. This helped initially; but we both continued to suffer from the effects of fatigue. I had the flight attendant call us every 15 minutes to ensure we were alert. Once we started our descent and approach into ZZZ3; our alertness increased. This is due to the increased task of preparing for the arrival. Total duty period was 11 hours and 8 minutes.I suggest the company not schedule more than one flight activity when spanning the hours of 0200-0500. When a red-eye is scheduled as the only activity; I do much better since I can rest all the way up to report time. Having to complete flying activity prior to the red-eye; coupled with a sit time in ZZZ2; is a recipe for disaster. Even after I had afforded all opportunity to rest prior to the event; I experienced fatigue during cruise. Had the effects degraded further than experienced; I would have had no choice but to divert to a location well short of our destination to get proper rest before continuing.