The team also announced they signed punter Dom Maggio to the practice squad. He was with the team during training camp and moved to injured reserve before the Falcons waived him in early September.
Morstead played seven games with the Jets this season after signing with the team Sept. 14. The Jets cut him Nov. 8 after Braden Mann returned from a knee injury that had sidelined him since Week 2. Morstead averaged 48.2 yards on 23 punts and dropped seven inside the 20.
Morstead spent 12 years with the Saints. He has averaged 46.5 yards per punt during his career, which includes Pro Bowl honors from the 2012 season.