The 19th century saw many names of baseball clubs in New Bedford. In my research I have noted roughly 60 different club names in New Bedford in a five year period. None of which I have found to have the name Sox included.
Here are a sample of 19th century area baseball team names. Teams were based in New Bedford unless noted otherwise:
Active Base Ball Club
Alaska Base Ball Club (Fairhaven)
Bay State Base Ball Club
Chicopee Base Ball Club
Clipper Base Ball Club
Eureka Base Ball Club
Franklin Base Ball Club
Independent Base Ball Club (Marion)
Ironsides Base Ball Club
Kit Kat Base Ball Club (Rochester)
Lowell Base Ball Club
Mattapoisett Base Ball Club (Mattapoisett)
Monohansett Base Ball Club (North Dighton)
Nautical Base Ball Club
Nobby Base Ball Club
Onward Base Ball Club
Red Rover Base Ball Club (Fairhaven)
Resolute Base Ball Club (Fall River)
Rising Sun Base Ball Club (later changed to Annawan Base Ball Club)
Riverside Baseball Club (Acushnet)
Rough and Ready Base Ball Club
Sons of the Ocean Base Ball Club
Star Base Ball Club
Wamsutta Base Ball Club
Young Tigers Base Ball Club
South Coast Vintage Base Ball 
Can you tell me more about the Riverside Baseball Club? I found an old stereograph of eight ball players that says Riverside B.B. club, taken in Topeka on the reverse side. They have an “R” on their shirts. There is no date but this looks to be circa 1880 to me.