I own a 1960 Schwinn Racer three speed that uses odd sized tires. Its time to replace them. I know I could get the new fitment tire for the bike... However Id like a little bit more width in the new tires.

Can I install 26x1.5 tires onto this bike vs. the 26 x 3/8 x 1-1/4 that is its originals?

No, you cannot, learned this long ago when I was ten and tried to fit a 2.125 tire onto a rim designed for 1 3/4". The "measured" size is different enough that one has to stay either in fractions or decimals. Sheldon Brown I think has an explainer on his site. There is another quirk to Schwinn tires, Schwinn used a rim with a hollow section and no "hook", the tires were designed to mate with this shape. So, typical tires of even the correct dimension may not seat as well as "schwinn" brand tires. The tires were made for them in their molds.

Even tires with fractional diameters can be different actual bead diameters too. Its really frustrating. Best thing to do is match the bead diameter (usually printed near the tire size, will be three digits, like a 559, 622, etc) and then worry about the width and all.

I understand now. Its the bead size. Old Schwinns use 597mm bead. Modern bikes use 559mm..... drats...

What is a "S6" rim. Can I put S7 tires on it?

I *think* you shoudl be ok but its best to consult Sheldon on this one.

https://www.sheldonbrown.com/tire-sizing.html

I can't remember if the problems I ran into were with 27" or 26" fractional sizes. I'm pretty sure it was 26", but read up on it to be sure.

Yes, the problem is with the 26". The Schwinn size was 597mm, English and most others were 590mm, and the modern mountain bike 26" are 559 mm. All 27" are 630mm.