This is the problem. You should always be able to comfortably make the full payment if you manage your budget proactively—I highly recommend YNAB for this. If you can't proactively manage your budget, you should be using a debit card.
If you pay the full balance on the due date, you get the interest-free loan. It's not exactly one month, but it's from purchase date until due date, which is something like 25-55 days.
You can pay your rent bill on August 1st, even if you don’t have the money for it until your paycheck on the 12th, and then once you get your paycheck, you can allocate the money appropriately so that you can pay off the bill at the end of the month worry-free.