I just thought of a variation for doing it with a hand drill, though still using using your drill press for the jig to get a precise center location:   
Drill a hole the size of the dowel in a block but not all the way through.  Without moving the block, then drill a 3/8"  hole in the center of the larger hole.   If you use a forstner bit for the larger hole, You can use the center point as a guide to drill the through hole and as long as you are careful and keep the drill square, the jig can be made entirely with a hand held drill.   Put this block over the dowel and use the 3/8" hole to guide the hand held drill.  

--Nathan, TX. Hire the lazy man. He may not do as much work but that's because he will find a better way.