OK. I know that the stages of a hand pump make each stroke more efficient, but what are these stages, exactly? How much more efficient is a 4 stage pump as opposed to a 3 stage pump?
In broad terms, stages act like a gear reduction to increase work done for effort put in.
Purely imaginary numbers...
Stage one takes in air at atmospheric pressure, say 200cc. Stage 1 pushes that volume of air into stage 2, which is 25cc in volume. The pressure increases 8x. Stage 2 pushes those 25cc(@8x pressure) into stage 3 @ 6.25 cc. That yields 4x(8x) pressure. Stage 3 pushes into stage 4 @ 3.125cc for another double of pressure. 8x4x2 = 64 fold increase of air pressure.
You may be able to pump 50 psi with a single stage pump. 50psi x 64 =3200 output pressure. Mechanical advantage. The penalty is increase pressure = increased heat and condensation.