HI Steve,
Standard, well tuned combo with active timer Axi2826/F2b (Kv=700), Spin 66, 11x5" 3 blade prop takes maximally 2000mAh from 6s2700 mAh battery during 5:15s flight. That prop gives little more power than 2 blade 12x6 APC. So if I recalculate those 2000mAh to 4 cell battery, it is 2000*6/4 = 3000mAh or less (because of lower power). So I would say you are in good numbers, even better than I would expect :- ))
However using more cells of the same capacity will give you longer flight, but you will fall to worse efficiency because of too high voltage for your motor, so it is not fully proportional. The secret is to match battery voltage, prop pitch and motor KV to minimize voltage headroom of battery voltage to necessary voltage to maintain wanted RPM. That head room is different (smaller) for constant RPM setup and different for active tomers. I guess your battery does not have enough voltage for safe power boost in verticals. Does it accelerates well? Do not you feel less power end of fligh when voltage drops? If yes you will need to change something, but I guess 5 cells will be already too much.