Why can one swim longer underwater by blowing out air?
This is purely a psychological response, and it does not help in decreasing the speed of CO2 level rise. As CO2 level in your blood rises, you instinctively want to breath out the 'bad' air and breath in fresh one. The latter is impossible under water, but the former eases you psychologically a little; you have "started doing something" with your predicament of high CO2 level.

Most freediving courses (as well as my own experience - humble as it is) suggest not to have the last breath substantially larger than your normal one; this comes from people who train to achieve 6-7 minutes dives, so I would assume they meticulously tried various methods.

The key here is not to have a victory over physiology; the key is psychology. Most people fail at diving because they unnecessarily get into a positive feedback loop: they get excited, their brain and muscles become even more active, they produce more CO2, brain receives information about rising CO2 in blood, and it becomes excited even more, produces even more CO2, etc, etc, until a panicked diver surfaces or inhales water. Without such unnecessary excitation, each moderately healthy human would be easily physiologically capable of 3-4 minute dives.

My take is the amount of air you have in your lungs should be primarily the amount that eases your psyche the most and helps to stop needless stress.