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Thank you for your question. It is my privilege to be able to help you.
Do you have an odor in between your periods as well, or only with your periods? How far apart are your periods when you are not on the pill?
Usually, when someone has a severe smell only with her period, it is because of old blood that has been in the vagina for a while and therefore became smelly. This can happen also if you have an infection in your uterus, which can cause the blood to smell bad as well. However, the usual vaginal infections that cause an odor, like bacterial vaginosis, would usually cause an odor even between your periods.
I would suggest that you try using a pad instead of a tampon to let the blood drain right away. This may help prevent the odor. If it doesn't help, you can also consider restarting the birth control pills, as the periods on birth control tend to be lighter and less likely to have blood build up and smell bad. If all this doesn't help, you should see your doctor to get cultures to make sure there is nothing else going on. Sometimes women with polyps or fibroids have heavy periods that can smell bad as well, so your doctor can check for these as well.
I hope you find this helpful. If you have any more questions please feel free to ask me.
Good Luck! Dr. SaulJE