Also Sunday, linebacker Brandon Bell (Mays Landing, Oakcrest) was signed to Cincinnati's practice squad.
Elliott was originally signed by the Packers as an undrafted free agent out of the University of Toledo in 2014.
Hill was one of only three quarterbacks to be claimed off waivers the day after final cuts occurred.
Hopes are that they will not repeat last season's terrible performance. The 6-foot-1, 220-pounder also had stints with New Orleans and Cleveland last season.
Green Bay made a decision to keep only two quarterbacks - Aaron Rodgers and Brett Hundley - on their active roster.
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Having lost out on Hill - this marked the second straight year the Saints claimed a Packers third-string quarterback, having nabbed Joe Callahan last October when the Packers cut him because they needed a roster spot - the Packers brought Callahan back on the practice squad after he cleared waivers.
Hill completed 14 of 20 passes for 149 yards and two touchdowns in three games while adding 71 rushing yards and a 23-yard touchdown run, the game-winning score in victory over Washington.
Bell, 22, enjoyed a solid preseason for the Bengals. A year ago, Elliott led the team with 13 special teams tackles and did record one sack.
Odom, an undrafted rookie out of Arkansas State, had 17 tackles and two sacks for the Falcons in the preseason.
In 2016, National Football League teams were required to pay practice roster players a minimum of US$6,900 per week, which worked out to US$117,300 over a full season.