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.