I have overwintered mine, but it's been in the house in a sunroom. I cut back the leaves and just water it periodically. I was told they need a period of dormancy, so don't worry if it doesn't flower over winter. It will come back fine next year. If you do the shed, is it heated? I think you need at least 45 degrees or so.

I have actually been to Logees in the winter, where they have dozens upon dozens of different kind of Jasmines..

They have a room there that stays very cool, to the point you can see fog from your mouth..
Some Jasmines do need a cool place to grow, to bring them to bloom. The Fragrant Olive love these conditions too..

Is there a way you can find out what kind of Jasmine you have...The hardy ones actually enjoy cool conditions..The tropical ones need to stay above a certain temp to stay healthy..

Good luck..I hope it works out for ya..