If you are paying for COBRA you are doing it wrong. You take those 60 days to find private insurance. You basically get 2 free months of health insurance since you cancel on that 60 day mark and never pay. If on day 58 you need health insurance you pay COBRA. Otherwise - your new private insurance kicks in on day 60.
[Yes! You could have history of heart attacks and not find cheaper private insurance but I am hoping that isn't you!]