“There wasn’t an obvious position for me,” Scott Amenta said.

Mr. Amenta, 30, was describing how five years ago he joined Spring, an e-commerce company so small that its employees could be counted on one hand.

Around that time, a mentor advised him to keep an eye on the nascent chiefs of staff in technology world. People in that position were seen as the next wave of founders, he was told. Reid Hoffman (the influential LinkedIn boss) had one! So did the Winklevoss twins!

So Mr. Amenta asked to be named chief of staff.

In 2016, after toiling patiently in business development, he got the coveted title, and took up a position that is suddenly everywhere. In the past half-decade, chiefs of staff have marched from the military and the halls of Congress into the technology world and beyond.

“There’s something about the title or the idea of the chief of staff that seems to be in the zeitgeist,” said Chris Whipple, the author of “The Gatekeepers,” a 2017 history of White House chiefs of staff. “I’m encountering more and more, not only in the political world but in the corporate world.”