>their underside is white
Au contraire. Ofc, I can't put a percentage how many cats don't have that. Besides there are full white cats out there.
Tho I could imagine something like a wingsuit, a hairy membrane straining between the front and back legs, allowing cats to glide over distances, like these guys: https://en.wikipedia.org/wiki/Flying_squirrel
I would guess cats lived on trees, stalking their prey above those heads among the branches and dropping on them even from a distance. This would also explain their ability to arrive onto their paws when they fall from a height.