You might want a 'Wald #940' (10" length) or 'Wald #945' (15" length). Steel. These have the Schwinn diameter for the seat tube and the modern diameter for the saddle clamp:

http://www.bikepartsusa.com/bikepart...ategory=search

http://www.bikepartsusa.com/bikepart...ategory=search

The pictures in the bikepartsusa catalog are inaccurate. Much better: download the Wald catalog --
https://waldsports.com/index.cfm/guides-downloads.html. I've ordered them from bikepartsusa and they look like the pictures in the Wald catalog, not the bikepartsusa pictures.

To re-emphasize: Your Schwinn saddle clamp with the narrow hole won't fit with this. The typical-size modern saddle clamps will fit.