Drill press slop

[bricolage]

I make 1/6th scale miniature chairs, so you can imagine that drilling small holes in the edge of thin wood can be a problem. When drilling a 1/16" hole, for example, I use a center drill initially to avoid the thin drill bit following the grain. However I notice that the center bit drills off center every now and then and found that there is slop in the quill or somewhere. I have a Ryobi and an old VIP Proline bench top drill press and they both have the same problem. I would hate to have to go buy a more expensive drill press to overcome this problem. Does anyone know of a way to tighten up said quill.

http://www.miniaturechairmaker.com