Car accident and rental car

Recently, I got into a car accident. Someone hit me from behind, which was not my fault. I opened a claim and request insurance to provide the rental car till my car got repaired. Since, I have a rental coverage in my personal insurance policy, Insurance people are saying I don’t need to have a separate rental car insurance. Here, I am kind of reluctant to use my personal insurance in the rental car. I am kind of scared (may be because of the recent accident) that if something happened to rental car, rental guys will claim and my premium go higher. Is this typical that we must use our personal insurance in this type of situation or we can ask for separate insurance for that repair period?