Given the information you provided, It is not such a big deal that you started the contraceptive pills after the end of your period, but it is a problem that you tossed away five of the birth control pills (BCPs). If you had them, you should have just continued taking the BCPs (and those 5 ones) as if nothing had happened and have an additional barrier

Use contraception (a condom with a spermicidal lining, for example) until your next period in 23 days. But now, you have to use the condom as an additional necessary contraceptive for 2 cycles (until the end of this cycle and during the next one). To be more precise, until the beginning of your second next menstrual period, that is 8 weeks from starting with BCPs, to be certain of the contraceptive effect. Combining 2-3 contraceptives (BCP condom spermicides) is always the safest alternative. A condom is also a wise choice in prevention of sexually transmitted diseases (STDs).