I'd play Rochester and Winona, as well as the new Somerby. Haven't seen the last one.
Rochester, if I've got the story right, was designed by Tillinghast because his daughter married a doctor in the area.
Winona is a great place. Definitely a true hidden gem.
I'm a little thrown off. Naccarato and Cosgrove mentioned courses in the Twins. You said you are up for a detour. Are you trying to go up there? If you have time for two or three rounds you really don't need to. Winona, Rochester, and Somerby will be a great trip. (Unless your visit coincides with the Nationwide Tour at Somerby.)
If you are trying to golf in the Twin Cities you really can't go wrong with the old-line private clubs like Minneapolis, Somerset, White Bear, Golden Valley, and North Oaks. And then there are some even more well known.