I sort of remember that cardboard band, now that you mention it. While I don't have a clear memory of it, I am pretty sure that the foam box was a "new" thing for BigMacs when it came out. Highschool years for me, I think, everything back then is turning into vague recollection.
Styrofoam Made Huntsman Rich - NYT - July 14, 2011
The Big Mac was introduced in 1968...the NYT article indicates Huntsman didn’t start his company until 1970. So the first clam shell box was likely Big Mac package.
Actually, I much preferred the Big Boy, but that was sit down dining and a slight bit more expensive.