Oiler Question

[JimF4M1s (Deceased)]

I have these originally marked oilers in my collection. Some say the SW on the cap instead of the bottom is an early Stanley Works for Winchester. Some have been found in what appeared to be original Winchesters.
But that has never been verified. It could have been for the M3 SMG "Grease Gun". And instead of Stanley Works, which in the consensus guess, it could be Star Watch Case company or Sparks Withington.