I insure my car through Winterthur with the Swiss equivalent of "fully comp".

Whenever I hire a car in the UK the hire company always lists a (to me) bewildering list of additional insurances I can purchase.

Does my Swiss insurance cover, in any way, hired vehicles?

Unless you have something very special then the answer is almost certainly 'No'.
If you insured a small car here in the lowest insurance group you couldn't expect it also to cover any car in another country, for example a top of the range Merc in England.