You can extend the Hex through a Short Rest without killing anyone, but you can't apply it to anyone else unless the original creature you applied it to has reached 0 hit points. And if you're getting them to 0 hit points but not killing them just so you can apply it to others later, I'd argue that this is even less intuitive than sacrificing small animals to your Patron when you wake up (and if you're worried about all the poor dead little squirrels, I'd point out you don't NEED to kill them, just drop them to 0 hit points :D)
To have both the spell slot back after the short rest, but also the spell being useful after it, the creature you Hexed will almost always have to first be killed.