A few years ago, the Greene County Jail, an imposing concrete building in Springfield, Missouri, ran out of space. Local police officers were arresting more people than the 601-bed structure could contain.

At first, Sheriff Jim Arnott, an upbeat man with a Missouri twang, tried to squeeze more beds into each jail cell. “You always put as many people as you can in a space,” Arnott told me. “You don’t ever have an adequate amount.” Eventually, the sheriff began to outsource inmates to neighboring j…