Peeps I've been to South Dakota 7 or 8 times and just wanted to throw a few things out there. Good advice from PBH his findings are spot on with mine.
When he said public has more cover than private, that is true in many instances. Especially if you're talking public that is government owned and not walk-in. On walk-in the landowners can do whatever they want, state/federal owned stuff doesn't typically get grazed. Huge difference.
And yes they move in the wide open. Its not like I watch mounters stroll across cattle pastures every year, but its a lot different than WI and MN. Around here the deer worry about danger spotting them before dark. Out there they can see their surroundings. They get up earlier in the evenings and go to bed later in the mornings.
Even in my pre-beast days I figured out that they lay wind to back on the leeward hillsides. Mulies do it too and they're 1/4 as smart as whitetails
If you see a buck heading to bed, know for a fact he's going to find a leeward hill.
The first buck I ever shot out there (rifle) was bedded in 1/4" tall grass in a heavily grazed cattle pasture. Use Beast tactics, but keep a very open mind.
If I were you I would cyber scout your but off and find the areas away from driving trails. If I know anything about North and South Dakotans, they love to hunt from their trucks.