Have you seen the Singer 401A dating information at
http://singer301.com/dating/default.html yet? It’s a Singer 301A site, but discusses the other slant needle machines made in Anderson, SC. The 401A dates are about half-way down the page.
Of the 10 Singer 401A machines that I’ve had (my family still owns 7 of them), I’ve found that the red S badge appears on the 1958 & 1959 machines, and not on the 1956 & 1957 ones. As far as I know, that’s the only significance of the red S badge, and didn’t know that they were more valuable.
What is the serial number of your machine? If it starts with “NB”, it’s probably 1958 or newer.
CD in Oklahoma