At the end of August, when your student plan ends, you will be offered COBRA coverage. You are not required to, and I suggest you do not elect coverage immediatly. You will have 60 days to elect COBRA coverage if needed, this is just your safety blanket.

Upon election, COBRA benefits are automatically retroactive back to the date your coverage ended. Since it is a 60-day enrollment period, this covers the entire time you will be without coverage. Plan all your doctors appointments to land in months you are covered. If an emergency does occur while you are uncovered, elect COBRA, pay the higher premiums, but be insured back to the date your student benefits ended.

Double check your benefit eligibility date to make sure you fall within the 60 days. While you will begin your new position in September, benefit plan coverage will normally not begin until the 1st of the month following start date. In some cases it won't begin until 30, 60, etc days after start. It is odd, but not unheard of, to get coverage day 1.