If you have started your hole JUST ENOUGH to get the drill in up to the edges of the lips,but not the whole body of the drill in,you can move the hole: Take a cold chisel and make several whacks radially around the side of the hole in the direction you want the drill to drift towards to get it back on location. The drill will drill more rapidly towards the chisel marks. Repeat till you are on course again. The chisel marks are arranged like an asterisk (*),except the marks are,as said,on the side of the hole you want to drift towards.
It is best if you check the hole for position before the cutting edges get too far in,or your hole will have a beveled side from making the drill drift away to 1 side. Drilling a hole the diameter of the web is a big help both for power needed and accuracy as stated above.
When you center punch for small drills,make sure the diameter of the punch mark is as wide as the web of the drill,or the drill can easily walk off center.
You can use a lathe center drill in your drill press to accurately start a hole. I do that all the time. It's like using a spotting drill,but most small shops don't have spotting drills on hand.