I found a new, old, recoil pad. It's in the original box, still has the screws and instructions with it.
I went to Google Earth and found 4311 W. Belmont Ave. in Chicago. A place called Kitchen & Bath 4 You occupies the space. I don't know if that building used to be the Red Head Brand headquarters or if it's newer.
There's an ad on page 47 of the September 1959 copy of Guns Magazine that references 4311 West Belmont Ave. About the only place I found the company was in old magazines and that's what took me down the rabbit hole.
As an example, here's the May 1930 copy of Outdoor America. The whole magazine is interesting, but look through it for the ads. I was going to say that one thing unifies them all. But then I did see one for down sleeping bags made in Canada.