Photo by Jeffrey Beall - Own work, CC BY 4.0, License
Except for the Missouri River, which is a massive lake/tailwater, all of the streams in South Dakota that sustain wild populations of trout are to be found in the Black Hills near the Wyoming border. These are the highest elevation mountains in the Continental US east of the Rockies, with their high point, Harney Peak, reaching 7,200 feet, nearly 4,000 feet above the prairie less than 20 miles east (Grizzly Bear Creek and Battle Creek drain the eastern slopes of Mt. Harney).
Included in the download zip file is a kmz of the streams designated by the SDDENR as "coldwater permanent fish life propagation waters" in the 2012 Surface Water Quality Assessment prepared pursuant to sections 305(b) of the Federal Water Pollution Control Act. In other words, these are places that will (or are supposed to) support populations of wild trout. Beware, however, that a fair number of streams listed here are designated as "impaired" or "threatened," generally because the streams are not meeting temperature standards. (In this era of global warming, such will be the fate of many trout streams). If you click on the stream segment in Google Earth, a bubble will appear which provides the underlying data set including the "Use Status" which will be listed as either "Good", "Threatened", or "Impaired".
Also included is a guide to Black Hills fishing prepared in 2002 by the SD Department of Game, Fish, and Parks. It may be a little dated, but it covers many of the streams in the dataset and will be useful if you're planning a trip.