But why? Because it's not even in standard time, except for around 1/60th of a time zone at best, if you're rounding to the minute.
If solar noon jumps from being at 11:35 in standard time, to at 12:35 under DST, at your coordinates, what does that matter?
Noon was at noon before the railroads. But ever since time zones were invented, that's no longer been the case.
Digits on a clock are just a number. If you care about when solar noon is, just memorize it.
The song is about a secretary who didn't get a lunch hour, so started an hour later than her boss.
Tech workers generally start at 9, but that started decades after the song came out.