Just replace your, ehm, drill bit, with some wire, like suggested. You can then even use your pin vice to hold the wire.
I usually use paperclips for this purpose; bend straight and cut off at the appropriate length. Superglue into the pre-drilled hole in the foot (or whatever) of your mini. When priming, I stick the wires of the minis into a piece of foam or pre-drilled wood. Then, when painting, I use a pin vice, or I stick the wire into a cork.
And when the painting is done, I cut the wire, so I can use it as a pin for basing...