Should The New Birth Control Pills Be Started Immediately?

due to a change in health insurance I am having to go to a new obgyn. My problem is that the first appt. that I could get is almost a week after my birth control pills will be finished ( i will be done with them on a saturday and the appointment is for the thursday after ward). I do not have any more refills so I must wait until I get a new perscription. I currently take Junel, which has a side affect of stopping monthly bleeding so I have not had a period for about 4 months. If I am able to refill my pills that day of the appointment can I start taking them asap?