I recently started messing around with TASing this game and noticed in WhiteHat's run that he doesn't know what causes the super jump at 3:40. The answer: it happens when you press jump one frame before landing on the ground OR if you make two frame perfect jump inputs. Just sharing the tech in case anyone decides to run this game.

EDIT: To clarify because I just realized that isn't TOTALLY right: the super jump happens when you're landing from a regular jump.

If you press jump one frame before landing from a DOUBLE jump, it instead skips the landing animation and lets you keep moving at full speed. Also if you press jump on the frame you land from a double jump, you skip the animation and immediately do a regular jump.