Smith tore his ACL on Oct. 23 and has been rehabbing this offseason. He is not expected to be ready for the Giants' offseason workouts this spring.
"Geno has a great, positive attitude, and he's looking forward to learning and competing in our QB room," general manager Jerry Reese said in a statement.
Smith's recovery period will put him at a disadvantage. He's going to compete with veteran Josh Johnson for the backup role behind Eli Manning, who has never missed a game since becoming the Giants' starter in November 2004.
Smith, 26, has plenty to prove. He has thrown just three touchdown passes in the past two seasons and is returning from a serious injury. The 2013 second-round pick out of West Virginia has just 28 touchdown passes -- compared to 36 interceptions -- in four seasons.
The hope is that Smith can benefit from working and learning behind Manning in a stable environment.
"I think the opportunity to learn from [quarterbacks] coach [Frank] Cignetti, coach [Ben] McAdoo, and to be alongside Eli and learn from him, a two-time Super Bowl champion and, in my opinion, a Hall of Fame quarterback," Smith said in the team statement. "I think it will work out tremendously for me. And it's also an opportunity to be on a real good team, a winning team, get healthy -- it's just a perfect scenario for me. I couldn't have written it up any better myself."
Smith's tenure with the Jets lacked stability. He is best remembered for his 2015 locker room altercation with IK Enemkpali. It was a career-altering moment for Smith, who went from starter to forgotten backup in a split second.
Enemkpali, claiming Smith owed him $600, fractured the quarterback's jaw with a punch to the face. Smith required surgery and missed nearly two months, losing his job to Ryan Fitzpatrick, who enjoyed a career year.
"Every day I'm pissed off until I get my job back," Smith told ESPN during last summer's training camp. "Until I'm a starting quarterback, I'm pissed off every day. That's my mentality, that's my competitive nature. I want to win so badly, deep inside of me. I'm not pissed off at anyone, but I do believe I'm a starting quarterback in this league, and I believe I can do great things."
Smith got another shot last season, but it was short-lived. Starting for the demoted Fitzpatrick in Week 7, Smith tore knee ligaments in the second quarter and was finished for the season. He underwent surgery in November, vowing to be ready for training camp.
Smith, who started as a rookie, compiled a 12-18 record in four seasons with the Jets. He struggled with turnovers, as he threw an interception every 24 pass attempts. He also was fined by the NFL for cursing at a fan at a home game and he received a torrent of criticism for missing a team meeting on a West Coast trip, claiming he got the time zones mixed up.
The Giants also signed cornerback and special-teams player Valentino Blake, who has played for the Jaguars, Steelers and Titans.