Osgood was presumably re-signed after Week 1 so that his contract was not guaranteed for the season. Osgood was with the team during the preseason and then was cut, seemingly to make it easier to cut Osgood later in the year if things don't work out or a better player is available.
Osgood has proven to be one of the best special teams players in the NFL over the course of his career, playing well in that role with the San Diego Chargers, Jacksonville Jaguars and Detroit Lions before joining the 49ers.
Osgood made the Pro Bowl in 2007, 2008 and 2010 as the AFC's special teams representative, getting All-Pro honors in 2007 and 2009. As a receiver, Osgood has recorded 44 catches for 749 yards and five touchdowns.
He was also honored in San Diego as a member of the Chargers' 50th Anniversary Team.