When the Kansas City Royals rode an otherworldly bullpen to consecutive World Series appearances in 2014 and 2015, the rest of baseball reacted the only way it knows how: by copying them. The three largest contracts ever awarded to relief pitchers were signed this winter, as teams look to imitate Kansas City’s model.

But what if the most important takeaway from the Royals had nothing to do with closers and setup men at all? A couple of teams located in opposite corners of the country believe they have found the real key: outfield...