I have just returned from my second year of attendance at a Badlands photography workshop, that is headquarted each year at the Circle View Guest Ranch just outside of Interior South Dakota which is one of the gateways to the Bandlands and just North of the White River. Doug Beasley of Minneapolis is the photographer who has been leading these outings for about 10 years. Phil and Amy kruse who are actual ranchers as well as proprietors of a wonderful Bed and Breakfast could provide you with some directions to wonderful settings that are on both public and private lands. Ansel Woodenknife is another good resource and anyone in Interior can direct you to his business (a thriving internet fry-bread business), and I'm sure he would offer advice and some history as well.
My suggestion would be to try to stay at least for a while closer to where the action is. Interior, S.D. with a population of 67 is an experience unto itself. Two bars-one of which makes great pizza (Wagon Wheel Bar) in a town without a restaurant is worth a visit as well as having the only gas close by-my recommendation is to fill your gas tank at every opportunity as well as having plenty of ice, water and snacks-this is real "outback" country. Gas stations are frequently without fuel. Real cowboys often ride their horses into the bar at night, and on most nights (before dark) a local brings a huge bull into the bar which patrons can sit atop if the mood moves them. I drove about 350 miles during the week on dirt and gravel roads without problems, but if it even appears remotely that it is about to rain, get back to pavement ASAP, as the roads turn to a glassy slickness you will want to avoid. A 4x4 is always a good idea, though on occasions some in our group drove sedans (with varying results). I've never met anyone in this area who wasn't very friendly, willing to visit and offer advice.)
If you want the postcard type of panorama, stay on the paved loop through the park, and if you are there early or late, or the light is good because of some of the amazing clouds that pass through you will rewarded, though I find photographing the badlands something of a challenge even with my panoramic camera.
Some of the various tables, (Wolf Table, Sheep Table, etc.,) provide incredible views and the Native Americans with whom we interacted were always very friendly and often stopped to chat. The area to the South and West of Scenic, S.D., takes you to some amazing grasslands on the way to a very far-reaching tabletop view of some beautiful country. You might find offerings left by Native Americans which are sometimes tied to trees and one should be very respectful of these.
Good luck, take more water than you think you might ever need, and you'll be grateful if it is kept very cold.