A year ago, someone touched my rented Twingo (AVIS) and I first noticed in the moment I returned the car; it was a small damage about 600 EUR.
The Lufthansa VISA Gold / FTL / Senator has an car insurance for rented cars, and they handled it without problem. They asked me to send a letter about the accident and the Letter, AVIS send to me. The money was during a week on my bank account (even before AVIS Charged my Credit Card).