The 32-year-old Branch returns to a Patriots team whose wide receiver roster has ballooned during the first couple of weeks of free agency.
Larsen, 28, late of the Denver Broncos, brings versatility to the squad, having played in all three phases of the game.
"He's played linebacker, he's played fullback, he's played in the kicking game. He hasn't played for us, so, we'll put him out there, see how it goes," head coach Bill Belichick said at the NFL's Annual Meeting earlier this week.
"Not sure exactly what his role will be. We'll give him a chance to compete. He's done a lot of good things that we like, that's why we signed him. We'll see how it goes. I don't know."
Fiammetta is a 25-year-old Syracuse product who spent his first two pro seasons with Carolina before going to Dallas last season.
He has started half the 34 games he's played in the NFL, rushing 11 times for 26 yards and catching eight passes for 73 yards.