Short-term auto insurance--does it exist?

Greetings,

I have been walking and bicycling a lot, and using my car less and 
less, and have begun to wonder why I pay for auto insurance each 
month. For the cost of my present insurance, I could more than 
afford to run my residual car-errands by taxi.

The irony is that living in the city enables a car-free existence, 
but is also a risk factor that raises your premiums!

I don't want to sell the car because a) I am attached to it, 
b) I make cross-country road trips now and then, and c) because 
I don't yet know how much I am going to like cycling for groceries 
when it's -40 this coming Minnesota February. 

Even if this state did not require liability insurance, I would
buy it anyway, because I'm like that. So just dropping my
coverage and taking chances with the law is not going to work
for me.

So: I really want to keep the car, and really insure it when
using it, but I don't have to be able to drive it every day,
with no notice. Just "one weekend a month, two weeks a year,"
so to speak.

Does anyone know of an insurer that will let me turn coverage
on and off with short notice, for short times? Sort of like 
renting my own car? Or one that offers a policy that covers 
you on alternate Sundays or something?

I am willing to maintain some sort of storage insurance, if 
necessary, to satisfy the companies' fixation on "continuity of 
coverage," without which you are over a barrel when you later try
to re-insure it.

My insurer and a couple others I've talked to won't touch it.
Hagerty's collector car policy would almost do it; my car is 
old enough, but they require that your primary vehicle must 
be another car, and insured by somebody else. Not your own
two feet! Not a bicycle! 

Is there really no middle way? Surely I'm not the first to ask
this, though my search on dejanews does not reveal it.

Cheers,

Phil Roban

proban at qwest dot net