I think people get confused as to what was low altitude and high altitude in the late 30s. in 1939 ANYTHING and EVERYTHING that did not have a turbo charger was "low altitude". There were NO workable 2 stage mechanical blowers in any country and the 2 speed single stage super charger was only 6-7 years old and the ONLY production engines with 2 speed superchargers were the British Armstrong Siddeley Tiger, Bristol Pegasus and Merlin X, The American Wright Cyclone (since 1937) and the P &W Twin Wasp (after the Cyclone), and the German inverted V-12s.
Given the state of the art in supercharger design (the actual impellers and housings) nobody was really ahead of anybody else at this point. Critical altitudes (full throttle heights) even for the 2 speed superchargers didn't go over 17,000ft and many of them were down around 12-14,000ft in high gear. In many cases the 2 speed supercharger was used to boost take-off power and climb out as much or more than it was used to improve altitude performance.
A good supercharger in the late 30s was capable of delivering a pressure ratio of about 2.3 to one installed in the aircraft. Really good ones were touching a pressure ratio of 2.8 to one. If your engine needed 44in of manifold pressure to make rated power then the critical altitude was going to be about 12,000ft with a 2.3:1 supercharger. Spinning it faster wasn't going to help, the design was maxed out. Spinning it slower would take less power to drive at lower altitudes, it would heat the intake air less and it would allow the throttle plate to open wider meaning less intake loss. A big boost in power at "LOW" altitude like the 3-7,000ft heights that ALL the production 2 speed engines used as their "LOW" altitude critical height.
A 2.8:1 supercharger could get the 44in MAP engine to about 17,000ft, however a 2.3:1 supercharger engine that needed only 39in of MAP could have a critical height of 15,000ft.
The supercharger on a Merlin III had a Pressure ratio of about 2.6:1.
Considering that the P-40 was "designed" (adapted from the P-36) in 1938 and production orders first place in the Spring of 1939 with the object of getting USABLE fighters into service as soon as possible with the least amount of development work needed there really wasn't much other choice. The Army was very interested in high altitude operations, they just were not possible given the currant technological state of the art.
As has been said before, Allison was a small company in 1937-38 with fewer than 25 employees in the engineering section. They also had no real production facility. In 1939-40-41 they knew that a better supercharger was needed ( work did start in 1938 on a two stage) but the rapid expansion and getting the basic engine into production sucked up a lot of time and effort. The Army was probably more at fault for wanting a variety of different engines than for not wanting a better supercharger. With the P-39 extension shaft engine, the Aircuda extension shaft pusher engine, the handed engines for the P-38 (and Aircuda?) and the V-3420 24cylinder engine the Allison staff may have been stretched way too thin to put proper effort in a better supercharger. With General electric supply the supercharger design and parts to ALL American engine makers until about 1937 there wasn't much of a pool of supercharger designers to draw from either.
Design got a lot better in 1941-42-43 but in many countries production of existing designs often trumped improved models.