"gone Winchester"

What does the term "gone Winchester" refer to, and what is it's origin?

A fighter pilot who runs out of ammuntion will say he has "gone Winchester" and needs to return to base and restock his guns.
Guess he needs to get a whole nine yards worth?

I do not know why they use Winchester to describe that state of affairs.

Winchester is a gun manufacturer. Possibly the pilot means he has no more military-quality ammo and is reduced to aiming a Winchester rifle.

Winchester also makes (or they did during the WW II) ammunition. Maybe it was a catch-word to mean "I have to go back for more ammo"?