I think we have been over this one a few times too.
During the 30s you had kind of a chicken and egg thing going on. In the early 30s you had 80 octane fuel and it just won't take much supercharging. Even 87 octane is only good for about 4-6lbs boost in most engines. Please remember that it is not only the boost but the rise in temperature due to the boosting that causes detonation ( along with some other things). Until you had better fuel you couldn't use a high pressure supercharger and since the superchargers that were available could provide all the boost required (or allowed) by the fuels being used there wasn't much research being done on improving them.
In the US this was compounded by General Electric having almost a monopoly on supercharger design. Wright, P W and Packard actually had small design staffs in the 1920s and early 30s and rather than spend time working on superchargers they ALL went to GE and bought not only supercharger designs but supercharger parts (like impellers) if not entire superchargers. I am talking about mechanical superchargers not turbos. Even some Auto racers ( Like Henry Miller) went to GE for advice on car superchargers. So basically you had
one supercharger design team in the entire United States and they had a few things wrong with their basic design. Nobody knew it because of the lousy fuel.
By 1937-38 87 octane was common, 91 octane was coming on the scene in America and 100 octane was being used in experiential batches and for record setting. With both Wright and P&W competing on the world scene and having larger design staffs than they had in the late 20s and early 30s they both realized that the GE superchargers left something to be desired and started designing their own. Accounts differ about Wright with either the R-2600 or the R-1800 G-200 series engines being the first to use a Wright designed supercharger (Kenneth Campbell). P W
might have been a little faster on the draw but not by much. P W then got into the 2 stage supercharger business.
Allison may have designed their supercharger in house but Allison during the 30s did a lot of sub-contract work for GE making supercharger parts (like impellers) as they sure weren't making any money selling engines until 1939.
The US learned a lot about superchargers and centrifugal compressors during WW II but it took awhile. The Supercharger on a Wright R-2600-8 as used in a Grumman Avenger only supplied about 7 to 7.5lbs of manifold pressure at 12,000ft compared to the 6-6.25lbs the Merlin III supercharger (Pre Hooker) would provide at 16,250ft.
You can have the "idea" of a two stage supercharger. Getting it to work is the big problem. Allison tried to go cheap on the first go round and use identical diameter impellers.