Flynn, 30, was released Aug. 10 by the New England Patriots, his fifth team. He has made more than $18 million in his career despite only seven starts and 17 touchdown passes.
The Jets, looking for an experienced backup for new starter Ryan Fitzpatrick, gave Flynn a physical exam last Friday and maintained the lines of communication.
The Jets made Flynn's addition official Wednesday.
Earlier Tuesday, general manager Mike Maccagnan said he planned to acquire a veteran at some point, adding, "I don't want to necessarily put a time frame on it."
Maccagnan wasted little time, giving the team some security at the position.
Aside from Fitzpatrick, the only quarterbacks on the roster are rookie Bryce Petty and undrafted rookie Jake Heaps. Petty, a fourth-round pick, is a lock to make the team, but he's not ready to play. By adding a veteran, the Jets probably will go into the season with four quarterbacks on the 53-man roster, counting Smith.
Placing Smith on short-term injured reserve is "a possibility" but "not likely," Maccagnan said. That would require Smith to sit the first eight games. Each team is allowed only one short-term IR designation.
Smith had surgery on Thursday to repair a fractured jaw after being punched in the face by then-teammate IK Enemkpali. The initial timetable for Smith was six to 10 weeks; now it likely will be eight weeks, based on Maccagnan's acknowledgement that Smith will miss the first four games.
Flynn, originally a seventh-round pick of the Green Bay Packers in 2008, attempted 16 passes last season in his second tour with the Packers. He has a 61.3 completion percentage in his career.
As for Smith, he still hasn't been cleared to be at the facility on a full-time basis, though he attended the team meeting on Tuesday. By the end of the week, he should be cleared by doctors to attend meetings throughout the day.
It might take two weeks before he's cleared for light physical activity, according to coach Todd Bowles.
Smith was reprimanded by team officials for playing catch with a friend on Saturday outside his apartment complex, disobeying an edict from the team. He wasn't disciplined, Bowles said.
It has been one week since the Jets made national headlines with the locker-room confrontation between Smith and Enemkpali, who was released immediately and was claimed on waivers by the Buffalo Bills. Bowles has instructed the players not to talk about the incident anymore with the media.
"That's not going to define who we are and what we are, going forward," Bowles said. "We've moved on."
Receiver Brandon Marshall, who backed Smith on national TV and radio shows on Monday, said he's done talking about the incident. He was the first player on the team to assert publicly that Smith "didn't do anything wrong," but he declined to comment on why he waited six days before sharing his opinion.
"If you don't want to ask me any questions about Geno that are football[-related], we're wasting our time," he told reporters. "Coach asked us all to move forward. I don't want to be a distraction to the team. I want you guys to respect that, and I want to respect Coach's wishes.
"There is a perception out there of me," he said. "Some of it is true, some of it isn't. I want to win, and I don't want my words to be a distraction to this team."