You can still have old blood come out even if it's a new period. Your vagina does its best at trying to clean itself out, but it's not always perfect. It's not abnormal to pass some old blood, or even some uterine lining, when you get your new cycle. If you're trying to avoid any "pregnancy scares", I'd suggest to get on some birth control or at least use condoms. If you're still concerned take a test or consult your doctor.