Yes, up to the limits of liability you have on your existing policy as shown in your declaration pages. If you are carrying comprehensive and collision coverage on your policy, those coverages will extend as well, with the loss being subject to the appropriate deductibles.
When you purchase the rental car company’s insurance be sure to understand what your coverages are and your responsibility in the event of a loss. One of my insured’s was surprised to find there was no vandalism or malicious mischief coverage available when the car was not in his immediate presence or control. He found this out when he left his rental car in the Ala Moana mall in Honolulu to go shopping and someone ran a key down the full length of his rental. Since he wasn’t there and had no control of the vehicle, the rental company insurance declined the request for payment and required him to make up the difference.
My clients seemed to be split on whether to purchase the rental insurance or provide the rental company with all the details of their private insurance including proof insurance was in force. My experience is the rental insurance is not cheap but if I do have a claim I can just turn it into them and be on my way.