Just started playing and I noticed its quite a bit faster to 'Punch Dash' and then jump at the end of the dash. It resets the punch and the momentum can be carried in mid-air. Makes that first "purify all the crystals" level FLY BY, so long as you keep control of your jumps and don't do a slam on accident.

Punch Dash > Jump > Touch Ground and Punch dash > Jump > repeat