If you're willing to modify the frame you could likely fit 40mm tires.
One way is to drill a hole in the driveside dropout and tap it for the derailleur hanger fixing bolt, which allows the axle to slide another 3/8" back to the very back of the dropout.
Another is to dimple the insides of the stays a bit more than they are, which is a bigger can of worms, as it often requires you to build or modify some tools. I have a highly modified C-clamp that I use for this that I made by cutting the ball end off a ball peen hammer, reshaping it to look more like a 'pill', brazing it to the fixed portion of the clamp, and then making a series of differently radiused wood blocks to support the other side. Generally I find that even if you bolt a hub in to support the frame when I do this, it still throws the frame out of alignment, and necessitates a realign, but most Chicago Schwinns are a bit wonky anyhow, which looks to be the case with yours too.
Panaracer makes some amazing 38c tires though, the file tread Gravel King come in a number of different colors, and they ride amazing. Only downside is that they aren't cheap, but they are high quality.