Hi! I Turned 40 Today. I Started Birth Control Pills

Hi! I turned 40 today. I started birth control pills for ovarian cyst, endo, and painful periods. I completed the first month and had to switch to a different prescription the second month. I was 2 days into the new prescription, having taking them all correctly not missing one dose when I had unprotected sex. I had been off my period for 2 days. Am I protected starting the new pack immediately?