No. Normally when the at-fault party's insurance company provides you with a rental car while your vehicle is being repaired, it does not include the cost of you taking out insurance with the rental car agency.
Property damage liability coverages of the at-fault driver typically will cover the repairs to your vehicle and a rental car when your vehicle is being repaired, due to the "loss of use" of your vehicle, but it doesn't normally pay for extras like insurance from the rental agency or an upgrade to the better or bigger rental vehicle type.
The reimbursement you will receive for the rental car is usually just a per day or per week amount that they stipulated to you that they would pay.
If you already have full coverage on the vehicle that is being repaired, then those coverages will normally extend to the rental car. This means that you won't have to take out car insurance with the rental agency. While most policies provide this coverage, we still recommend that you check with your car insurance company to find out what exactly your car insurance policy covers with regards to rental cars.
Also, if you put the rental car on your own credit card (because the insurance company will reimburse you with a check at the end of the rental period), then check to see if your card offers rental car insurance coverage as part of your benefits. When this benefit is offered, insurance coverage is normally secondary to your own car insurance coverage.
After talking to your credit card company and personal car insurance agent, you should be able to determine if you need extra coverage from the rental agency. Likely, you will have enough coverage through your personal policy and credit card benefits that you won't need to get extra insurance from the rental agency.
Remember, if you do decide that you need this coverage, then it will be paid out of your own pocket and not reimbursed by the at-fault driver's insurance company.
If your agent is not able to explain to you what to do in this situation or fulfill your car insurance needs, contact a CarInsurance.com agent and they can explain not only insurance terms and explain claim procedures but also tell you how they can get you low cost car insurance.