The simple answer is: Yes. Your insurance policy will cover your rental car as long as you have comprehensive and collision (full coverage) on your current policy. Simply put, whatever coverage you have on your current policy, it will transfer to a rental car.
Here is where things get tricky. If you're renting a car from a rental car agency, and that agency is charging you, or the insurance company a daily rate, then you could be on the hook for that daily rate if the car is in the shop being fixed by your insurance policy.
If you're in a new place, or if you have the rental car for a longer period of time, it's our suggestion that you purchase the insurance agreement from the rental car agency. This will make sure that the daily charges are paid for by the rental car insurance policy.