I just noticed that in the world record run (by Gampling Frog) James Pond doesn't jump three times after finishing the first level. Is there any trick to avoid this?

The closest I've come to is holding jump and stretch just after it counts the score until after he does one jump he stretches, he doesn't do the other two jumps.

EDIT: And holding right and jump gets him moving after a single jump, can't work out how to cancel it all completely though.

I lucked into finding out how this works in a recent run.

To get 0 jumps every time, you must DUCK while going into the barber pole to end a level. That places Pond in the crouch animation. Then holding right and jump on exit will let you jump straight away.