The annual Atlas Van Lines moving survey gives an estimate of each state's net migration. South Dakota beat Minnesota in the ratio of folks moving in versus folks moving out, but both states lost more than they gained:

Atlas Van Lines moving survey 2013

(click to enlarge!)

In Minnesota, 58% of Atlas Van customers were moving out. In South Dakota, 52% were outbound. North Dakota enjoyed the highest in-migration rate, with 67% of moving van customers coming in, likely to the Bakken oil fields.

Even if a big chunk of those dissastisfied Minnesotans are coming to South Dakota, it's not quite enough to stop the bleeding.