In reply to Roger Burrows:
I made the same discovery a number of years ago with about 10 days to go before my holiday departure. There was something of a panic. My then insurance company - can't remember who it was, as only an idiot sticks with same company in these days of loyalty penalties - replied in the same vein as yours. It was a case of no, we don't insure you outside the EU, no, you can't pay an extra premium to get insured. Believe it or not, they suggested we arrive at the Swiss border and arrange something then! Needless, to say we ignored this suggestion. Instead, I spent several hours telephoning companies till I found one that included Swiss cover. Again, can't remember who it was. It meant cancelling my then current insurance for which of course I had to pay.
I haven't taken a car to Switzerland for a few years now and we only take our van when going abroad and we insure that through the Camping & Caravanning Club. I am under the impression they are more sensible. I don't know if they cover cars as well, but it could be worth a try. You don't need to be a member.
I am sure the reason for this problem is that as Switzerland isn't an EU member they don't partake of some special insurance arrangement provided by that much-maligned generator of red tape, the Brussels commission, so dealing with claims in Switzerland will cost them a lot more and therefore result in their being less competitive in the motor insurance market place.
All this makes you wonder how many Brits blithely travel through Switzerland totally unaware that they have no insurance cover. It is also surprising to me that the Swiss border officials don't demand to see proof of cover as used to be the case all over Europe in the days of the old Green Card. Remember those?
Not sure this is going to help you very much, but hope you've got a day or two spare before you leave. Good luck!