No, you are not required to use the mileage method for a rental car just because you use it for your personal vehicle.
The IRS allows you to choose the deduction method independently for each vehicle you use, for business purposes.
For short-term car rentals, you generally must use the actual expense method because the standard mileage rate is reserved for vehicles you own or lease long-term. This means you should deduct the specific costs of the rental fee, gas, and tolls rather than tracking miles. You can continue using the standard mileage rate for your personal car while simultaneously deducting actual expenses for your rentals.