Will this mess up my cycles

So i have been on birth control for a long time already like a year and I finished my last pack then got my period june 26 and finished july 1 then i had to make an appt for a pap smear in order to get my new refills for my birth control but the cranky lady in the front desk at the clinic said i cant make an appt until after the 7th or 8th cause my doctor is gonna be out so will waiting until then to actually get an appt and then waiting idk how many more days to get my prescription throw my cycle off and make me be unprotected when i start a new pack again a week or two after finishing my period? Because i think thats how long its gonna be until i can actually get them, its stressing me out