To get truly quick UPS, you need to become stiff. Proper vertical jump training will help you do this.
I'm not talking about being inflexible that won't help you at all. You need to have a great deal of muscular stiffness to lock up your muscles quickly and reverse the movement fast at the bottom of your jump.
To become stiff, you need a lot of strength in your lower body and you need to be able to use that strength very quickly.
Three things you need to do to become stiff are:
First, get your feet strong
To do this, train barefoot or as close to barefoot as possible and you also need to be conscious of how you walk. When you walk, be sure to spread your toes and grip the ground.
Second, get your ankles strong and your plantar flexors stiff. An easy way to do this is to jump rope and do line hops before your exercise. Do your line hops with one and two legs, forwards and backwards over a line and also laterally.