Difference between the 38 and the 39/40 transmissions is that the 39/40 have a lengthwise mounting surface and the housing/shaft is longer, (and fatter from what I see), While the 38 transmission has a shorter housing/shaft and an opposed/crosswise mounting surface.
And the 39/40 has the little screw on the front face that retains the input shaft bearing; which necessitates having the correct 39/40 bell housing which has the little dimple to clear the retaining screw.
So a 39 or 49 Cadillac/LaSalle transmission should bolt right into a 1940 car, but a 38 trans. will not. A 38 side shift transmission is unique to it's own year.
If you are still interested I have a 1938 LaSalle transmission and could check for the number(s) you were looking for.
Regards,
Bob, CLC 24870