Thomas spent the 2008 and '09 seasons with Miami and part of the 2010 campaign with Detroit. He started 12 games for the Dolphins in 2009 and was part of the Lions' final cuts earlier this month.
The Patriots lost center Dan Koppen to an ankle injury in Monday's opener against Miami and were without lineman Sebastian Vollmer because of a back injury. Dan Connolly moved to center in Koppen's absence.
Welch was added to the Patriots' 53-man roster just prior to the opener and played on special teams against the Dolphins.
The Patriots also added tight end Garrett Mills and linebacker Jeff Tarpinian to the practice squad while releasing linebacker Aaron Lavarias from the practice squad.