If you aren't making a big change in hole size you can simply hold the prop in one hand, drill motor in the other and enlarge the hole..."freehand".

Drill from the back side. I fly a lot of small, oddball sized engines and keep a small drill index in my tool box. I hold the drill bit steady with a pair of pliers and forcefully turn the prop down onto the drill bit.
I don't recall any props that didn't run smooth enough afterwards....but I'm usually only using small $2 props this way.

For a large [expensive] prop, chuck a drill bit into your drill press that is an exact fit in the prop's existing hole...then use that arrangment to position your prop. You will need to come up with your own clever method of clamping the prop in place.

A drill press operation not set up correctly can introduce an off center / cocked hole just as easily if not more so than someone who is using a cordless drill with the prop in his free hand. If the change in hole size isn't too great, the bit will track perfectly through the prop without any outside influences.