GtbPappy,
Don't think I can help with your serial # request . . . the closest serial # I found to your #57388 in my SRS Research books is #57350 - a Spencer Carbine issued to Company E, 6th Indiana Volunteer Cavalry in July 1865. Maybe it was issued to the same unit . . . really hard to be definite. Spencer Carbines were not issued in any really logical sequence of serial numbers. More of a "need" basis than anything else.
Sorry, best I can give you.
Two Flints