There were two categories of players established once the Topps and Bowman suits started being decided: exclusive players and non-exclusive. I forget the year without checking my notes (1954 I think) but at one point Bowman had hundred of exclusive players compared to a few dozen for Topps with a handful, mostly stars, having non-exclusive deals. So Topps started signing prospects and using coaches and managers to round out their sets. It was unrelated to where each company was domiciled.
I have a section on all of this in my new book:
http://www.scribd.com/doc/126643197/...m-1938-to-1956
It's a free download so no worries!