No, a car hire excess insurance policy runs independently of anything involving the rental company you may be using. In most cases claims for the original car rental excess will be made against the renter’s credit card as per the deposit left, these costs can then be claimed against the independent car excess protection policy.
The car rental agency will of course still offer you their own insurance products at the desk as they make more money on the extras being sold. Should you have an independent car excess waiver insurance policy you know in advance what is covered and what you may potentially still need cover for and can take this into account.
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