Credit cards that carry a balance almost always charge for new fees on an effectively daily basis.
Credit cards that do not carry a balance (are fully paid off by the statement due date) almost always do not charge interest.
If you are able to fully pay off your credit card for the period leading up to the new statement being issued, but unable to pay it off if you added on the new charge, then charging the cost after the statement comes out gives you an interest grace period.
If, however, you are already carrying a balance, then the "statement date" doesn't have a discontinuous impact on the total interest you'll be charged. The effect of delaying an additional debt will cause a reduction in debt carrying cost (interest), but the specific statement date will just mostly just defer when you are told about the interest on the new debt, and have a very small impact on the amount of total interest you'll owe on that debt.
You know you'll own 100$ on a credit card this month. Last month you paid it off in full, and have a new 1000$ cost. Your credit card charges 20% annual interest (1.5% per month, 0.04% per day).
You cannot afford to pay off the 1100$ when your next statement is due, but you can on the statement after that.
If you charged before your next statement, then carried it for a month, then paid it off, you'd pay a total of 1115$.
If you charged it after your next statement, paid off the 100$ on the next statement, then 1000$ on the one after that, you'd pay a total of 1100$.
There is a discontinuity here of 15$ from moving the date of debt by a day or two, far greater than the interest on 1000$ over 2 days (which is about 80 cents).
Now, suppose you have a 100$ credit card debt you didn't pay off last month. You'll owe 101.5$ next statement if you don't charge. If you charge the 1000$ the day before the statement, you'll owe 101.5$ + 1000$ + 0.4 on your next statement. Then another month passes, and you owe a total of $1118.43.
If you instead charged the 1000$ the day after, you'd owe 101.5$ on your next statement. Then you'd owe 1117.62$ on the statement after that.
The difference -- 80 cents -- is basically 2 days of interest on 1000$. There is no "discontinuity" due to the statement date here.
Note that the terms and conditions of your specific credit card can be different than what I modeled above.