Just to point out, if the damage is EUR3001, you're still on the hook for the first 3,000.

Companies sell stand-alone insurance for these situations. Have had an annual policy from insurance4carhire in the past. Less than £50, but they do daily for just a few pounds, too. That takes it to a zero excess and they also cover lost keys, tyres & chipped windscreens which the major hire firms leave you on the hook for, too (supposedly).

I've never claimed through these guys but a friend has & tells me it was absolutely seamless. I think this is also the company that Amex provide in the bundle with a Platinum card.