Friday's move ends Stafford's season with two games remaining.
He had played in only three games this season and 10 last year because of injuries. Stafford's shoulder was separated in the season's opener at Chicago. He later aggravated the injury.
Coach Jim Schwartz has not announced who will start Sunday at Miami.
Stanton has led the team to two straight wins, but hurt his non-throwing shoulder last week against Tampa Bay.
Hill broke his right index finger Nov. 25 against New England and missed the next three games.