I have an old house and want to install a range hood. Can an under-cabinet range hood be converted to one that will vent through the wall?

Here is the set-up in my house: stove, a window immediately above that (about 20" top to bottom), then, about 12" above the window, a duct for a fan (or hood?). The duct vents directly out the wall and used to contain a fan. Immediately above the 8" round duct is the ceiling (and I mean "immediately"--there is about 1 mm of space between the top of the outlet and the ceiling). This is why I am hoping I can somehow convert an under-cabinet range hood.

My other question is, is it better to vent through the wall or through the roof. I live in Michigan.

Thanks for any help you can provide.