Can I use narrower 32-597 tires on Schwinn S-6 rims?
As far as I can tell, the only available tires for the S-6 rims used on Schwinn three-speed and single-speed roadsters like the Speedster and Racer are the Kenda K23. I have been unhappy with them. No matter what I have tried, I have had a devil of a time getting them to seat properly. I think I have maybe a 25% success rate. I was reminded of this again this morning as I pedaled to the train station and felt that lumpy ride in my butt.
So, I'm interested in alternatives. The Kendas are nominally 37-597 tires. There are Raleigh (CST?) and Schwalbe tires available on eBay (and likely other online sources) in 32-597 size, that is same diameter, but I think a fair bit narrower. They often come from Europe though, so the cost to test them out is kind of steep. Should I be able to use these tires (get them to seat properly) on the Schwinn rims?
Schwinn Speedsters and Racers are very common here in the Chicago area. I am on the verge of giving up on them as my beater bike of choice because of this one problem. (I could probably switch to 590 rims and get the brakes to work with a rat tail file. That would probably cost more than these bikes are worth though - rims+spokes+tires)
Yes, as long as the diameter is 597 mm the tires should fit. But finding 26 x 1 1/4 tires, especially good ones, is not easy; and if as you say they have to come from Europe, they're not going to be cheap.
If you're worried about it as a gamble, concerned that they might not fit, I wouldn't lose much sleep over that. There are enough people looking for good tires in that size that if you're not happy with them you can probably unload them through this forum (offer them for trade) or resell through ebay at little or no loss, perhaps even a profit.
Switching to 590's is not a bad option, especially if you can build your own wheels (Sun CR18's are nice lightweight rims) or get a pair off a used bike, either English or Japanese. As you say, there are a lot more tire options in that size.
Those tires may also not work well with the Schwinn straight sided rims. Most newer tires are designed for hook bead rims as are the ones you mentioned. They will blow off the Schwinn rim. Roger