Saturday's moves come as the Jets prepare to face the San Diego Chargers at MetLife Stadium on Sunday.
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Payne dislocated his left wrist during training camp and had surgery, but remained on the Jets' active roster because the team hoped he could play at some point this season.
Tevaseu has bounced between the Jets' active roster and practice squad this season. He provides insurance if defensive end Mike DeVito, listed as questionable with a sore knee, is unable to play. Rookie Kenrick Ellis would take DeVito's starting spot if the veteran defensive lineman is inactive.