I experimented with this a little bit more, and it also works if you're riding a carpet, get the Pidget to respawn, and pick the other Pidget off its carpet while you're riding yours.
Edit: Demonstration of the trick on 1-2:
http://dehacked.2y.net/microstorage.php/info/197600381/Pidget21.fm2
Unfortunately it doesn't affect your vertical speed at all, so it's useless in 3-1.
Edit 2: I experimented more, and apparently, if you go through a door while riding a carpet, your feet freeze in place
for the rest of the game! You can't walk or duck anymore, you have to get around by jumping.
So I'm guessing there's a "carpet riding" flag which gets accidentally left on in that case. So the mach carpet trick would be a case of unsetting that flag while you're still on the carpet, making the game think you're on solid ground instead.