Can you change the day to start a new pack of birth-control?

Yes, you can. You do so by starting the new pack before you've been through all the placebos, or "sugar" pills. For instance if you're a Sunday start, and want to start the pills on Wednesday, you'll finish the active pills in your current pack. Then take a placebo pill on Sunday, Monday, and Tuesday of your period week. On Wednesday, you'll start the new pack. (If you're worried about how the day of the week labels on the pills are going to work, and can't figure it out, just bring the pack in to your pharmacy and they'll show you how to relabel them.)

The important principle is to never have more than seven pill-free days. Fewer is OK.