The number "3" on the back appears original. There is no #3 listed on the Dodgers' 1960 Roster, so it was likely an unused Jersey or one from Spring Training (from a player who failed to make the team).
If it was a rostered Player, there would likely be evidence of an embroidered name in the area of the tail tagging. So it's probably just an extra Jersey that was not utilized during the regular season.
Still a very neat and attractive Flannel