The Patriots assistant had not been considered a top target, but he quickly emerged as the Giants’ choice. Matt Rhule, presumed to be the leading candidate, agreed to a deal with the Carolina Panthers.

The Giants, who joined the N.F.L. in 1925, have become known for a staid, prudent management style. True to form, on Tuesday morning, the team was expected to do something predictable: Hire Matt Rhule, the former Giants assistant coach who has become a rising star after rebuilding downtrodden college football programs at Temple and Baylor.

But early Tuesday, Rhule jolted the N.F.L. coaching carousel by agreeing to coach the Carolina Panthers. Roughly an hour later, the normally judicious Giants had some thunderbolt news of their own: The 19th coach in the team’s 95-year history would be Joe Judge, a special teams coordinator and wide receivers coach for the New England Patriots who has no head coaching experience at any level.

The exact terms of Judge’s contract have not been finalized, but an announcement from the team was considered imminent, perhaps as soon as Wednesday, according to a person with knowledge of the team’s decision who requested anonymity.

Judge, 38, has been a part of three of the Patriots’ Super Bowl championship teams and was on Coach Nick Saban’s staff during two of the University of Alabama’s recent national championship seasons. Before Judge interviewed with the Giants on Monday, team officials had reached out to Patriots Coach Bill Belichick, a former Giants assistant, who gave Judge a glowing recommendation.

Though Judge’s background with the Patriots is on special teams — an assignment Belichick once held with the Giants — Belichick told the Giants that Judge could coach any facet of football, adding that the two had frequent conversations about offense and defense.

Still, Judge’s hiring startled the pro football community, because the Giants had scheduled an interview with Josh McDaniels, Belichick’s more prominent and sought-after offensive coordinator, who was the head coach of the Denver Broncos in 2009 and ’10. The Giants had interviewed McDaniels after the 2017 season before hiring Pat Shurmur, who was dismissed last week after a 4-12 season. Two years ago, McDaniels agreed to coach the Indianapolis Colts but pulled out of the job to remain with the Patriots.

Last week, the Giants interviewed the Kansas City Chiefs’ offensive coordinator, Eric Bieniemy; the Baltimore Ravens’ defensive coordinator, Don Martindale; the former Green Bay Packers Coach Mike McCarthy; and the Dallas Cowboys assistant Kris Richard.

McCarthy, who won a Super Bowl with the Packers, seemed a good fit for the Giants, but he was lured away over the weekend by Dallas Cowboys owner Jerry Jones. On Tuesday when Rhule accepted a gargantuan deal from Carolina reportedly worth $60 million over seven years, it appeared that the Giants had been outfoxed in a race for the two most recognizable candidates.

There was also the possibility that the Giants’ current and longstanding management structure — in which the coach reports to a general manager, in this case Dave Gettleman — had become an impediment to attracting a new coach. Increasingly, teams have begun allowing head coaches to handle many personnel matters once reserved for general managers.

Jed Hughes, a leading coaching and executive search consultant for the sports industry, said on Tuesday that the momentum of the N.F.L. was carrying it toward a model in which the head coach runs the football operations.
“Having the head coach in control is important,” Hughes said. “The coach has the ability to create the culture.”

It is not clear whether Judge, especially given his lack of head coaching experience, will be granted any added off-field duties with the Giants. In a news conference last week, the team’s co-owner John Mara, speaking alongside his ownership partner, Steve Tisch, conceded that he would consider altering the power dynamic within the organization.

“I’m always willing to look at whatever’s going to improve the team,” Mara said when asked about making such changes, “and if I felt that there was somebody coming in here as a head coach who wanted a different role and he could convince Steve and I that that would make sense for our organization, we would certainly consider that.”

Whatever Judge’s specific role will be, the Giants acted swiftly after the news that Rhule would be going to Carolina, agreeing to hire Judge by midday, although ESPN reported that the Giants and Judge began negotiating contract terms on Monday night.

The Giants had no comment on Judge’s hiring Tuesday, but it’s possible they saw an inspiring example in one of this season’s strongest teams. A leading candidate for the Coach of the Year Award in the N.F.L. this season is Baltimore’s John Harbaugh, a former special teams coach. Harbaugh won the Super Bowl seven years ago with the Ravens, who have boldly built their current offense around the second-year quarterback Lamar Jackson. They are the top-seeded team in the A.F.C., and Jackson is the prohibitive favorite to win the league’s Most Valuable Player Award.

But Giants fans could be forgiven if Judge’s hiring resurrects some uncomfortable recent memories. When Ben McAdoo was named the successor to Giants Coach Tom Coughlin before the 2016 season, he had been a longtime assistant and coordinator, but it was widely noted that he had never been a head coach at any level of football. He lasted less than two full seasons, departing with a 13-15 record.

A native of Philadelphia, Judge attended Mississippi State, where he played a variety of positions. After graduating in 2005, he became a graduate assistant with his alma mater, before moving to Birmingham-Southern as a linebackers coach. He then became an analyst at Alabama for three seasons under Saban, who is a close friend of Belichick’s. Judge joined the Patriots in 2012.

Matthew Futterman contributed reporting.