Smith rolled into Cioffi, and he injured his leg, coach Rob Chudzinski said.
Chudzinski also said he had reached out to wide receiver Davone Bess, who is out for what the team says is "personal reasons."
Chudzinski said Bess was dealing with a family issue and the team would give him all the support he needs, but declined to provide any details.
In other injury news, tight end Jordan Cameron remains in the NFL concussion protocols, while guard Jason Pinkston and defensive lineman Phil Taylor both suffered concussions in the loss to the Jets and will be evaluated.
Cornerback Joe Haden aggravated his bruised hip and is day to day.