FLORHAM PARK, N.J.—Before practice earlier this week at the Jets’ training facility, receiver Brandon Marshall and running back Matt Forte pored over a sheet taped to the wall outside the locker room. It wasn’t a practice schedule or a diagram of plays. It was a bracket for the team’s annual ping-pong tournament.

As the Jets prepare for the coming 2016 season here in Florham Park, there’s no shortage of leisure activities to break up all the football. There are computers, a pop-shot basketball hoop and a pool table in the...