Bruce,
Ruth indicates S.G. used free-issue barrels (Underwood & others) during production clean-up, so I think you are probably correct. However, I'm not sure there was an Underwood barrel on my receiver when I got it.
I guess I'm looking for SAGINAW S.G. data in the early S/N block where S'G' made the receiver, marked it SAGINAW S.G. (Ruth page 433, example S/N 1825223), and sent it to the Saginaw plant for assembly.