Crewdog...
Welcome to the club, oh, Fellow Garanditis Sufferer! And let us know she shoots!
A bit of a minor point, first. If your rifle has a Springfield receiver and Harrington & Richardson barrel, I'd think the barrel's not original. I'm not saying it couldn't be, but it's rather more than a little doubtful.
As far as determining MAP status -- yes or no, that is -- my understanding is that such paperwork has long since been discarded. It might be possible to get a more-probable-than not determination based on the serial number.
Maybe... a loooong shot maybe. Hard documentation showing your 5.8 million serial was sent to Outer Innerstan in 1960? Nope. (Sorry to rain on your parade!)
From the way you phrased your post, I am assuming that you obtained your rifle through the Civilian Marksmanship Program. If so, CMP
might be willing to satisfy your curiosity verbally, but I don't think they'd want to document anything. (Though it wouldn't hurt to ask; the worst the CMP can say is "No!") (Well... they might say "Hell, No!")
Very limited Unit of Assignment lists were available at one time from an outfit called
Springfield Research, (SR) which has since gone out of business / inactive. At best, the service was pretty much of a crap shoot; the chances were pretty slight that any particular serial might be included. You may be able to get at the old SR data through
Springfield Research Service serial number search start here, and following the instructions given there.
HTH
Ben Hartley