You mention "the Dakotas" as an option in your opening post and that is still your best bet along with Nebraska. South Dakota will probably be the closest in actual driving distance/time but North Dakota and Nebraska will have similar hunting but likely cheaper due to being somewhat less known.
One might "stumble into" decent pheasant hunting in MN but one is playing some long odds now days. The western part of the state is the best with hunting being better the closer to the Dakotas you get. A good rule of thumb is one will average two reasonable opportunities on roosters per day in MN. That is not two birds in the bag, that is seeing two roosters inside 40 yards per day. It is also considering most to all hunting done on public land as farming is pretty intensive with little cover left and what decent cover there is is often leased out to others or reserved for friends and family.
Any place in MN providing Dakota-like hunting will be a game farm and likely an expensive one. You can do much the same closer to home for a lot cheaper. Compared to WI, MN is a pheasant hunting Mecca but compared to places a little further west MN is WI.
