Yeah, it's doubled alright yet, I must have mistakenly scrolled up when citing Opalescent red, plus, I'd thought sure I'd copied Flamboyant red in 1959 earlier off the catalog too, but nope both are in error. So, my credibility in this topic is sucking. [grin]
However, Your question of 'how may different reds' got me thinking, 'Where there 3?' And which year in the catalogs are each color cited?" There's only 2 different reds, the earliest, although it's possible the phantom red color could have differences in blends that appear a little orange-ish, and finally; the latest 1959's Radiant Red. According to the catalogs is the only year Radiant red was on a Phantom. Which, incidentally, is why I'd first thought yours was a 1959 because the red on your bike is not the old standby it's like Opal, Flamboyant or Radiant. Schwinn called that a 4 coat process; metal coating, red lead primer, a silver or aluminum and finally transparent red, green or blue, etc. I can only guess that, the differences between Opal , flamboyant and radiant is the tint in the final coat.
That could only mean, your bike is a 1958. That is, your serial numbers, given the 'Radiant' paint would be November 26 and 27: 1957:
11/13 ------------------ S30503 -------------------- S37123
11/14 ------------------ 7E82769 ----------------- 7E86306
11/19 to 11/25 ------- 7E86307 ----------------- 7E99655
11/26 to 11/27 ------- S37124 -------------------- S40029
Most likely, that frame was stamped on Nov. 26, 57 and delivered to bike shops early 58. I.E. if there is a 1958 catalog then, we'd know it's not a 1 year only color but 2: 58 and 9.
And your badge, a very special one for the 1st appearance of a Phantom's special or newest color? Maybe, likely, given its rarity; probably.