Ouch...sticks out like a sore thumb.

Believe it or not, I feel your pain. It's a lot easier to run the 1/16" drill through a piece of 1/2 or 3/4" dowel rod roughly 1/2" long. Stop drilling when the bit is sticking out of the dowel roughly 1/4" and cut the shank off even with the surface of the piece of dowel. A little dab of epoxy on top keeps the bit in place. Makes the bit less wobbly and gives you more control. Dad was a show off and made his out of knurled brass, but he was a machinist. Also a handy tool for drilling drain holes in headlights that have a condensation problem.