The tires are 26 x 2.125, so a ballooner.
My comment about ballooners had more to do with identifying the frame. When Schwinn started making middleweight frames, they narrowed the frame at the rear. There is supposed to be some way to tell by how many fingers you can get in at the seat stay bridge. I can't remember how many, but it did change for sure. So that is one thing to nail down. Balloon tires and rims will fit the middleweight frame, barely.
Also, Schwinn's middleweight rims (S7, 26x1-3/4") will not take balloon (S2, 26x2.125) or non-Schwinn middleweight (26x1.75") tires. So, if the frame is middleweight, then the balloon rims cannot be original.
Well... maybe one exception. When Schwinn brought Balloon tires back in the late 70s did they widen the frame back out or did they just use the middleweight one? I can't remember.
Another thing about the frame that changed from 58-59 was how high the seatstays are, so I've been told, causing a change in the fenders. Does anyone have the measurements? If it happens to be the later version, and is also 61 or earlier as outlined in GTs58's post, that would narrow it down to 1959-61.
EDIT: I just put my tape on TOP of the chainstay right at the bottom bracket and measured to the TOP of the seatstay right at the back of the seat tube. That is effectively the same as a center-to-center measurement. I measured 16-1/8". Bike is a 61 Speedster (S7 middleweight).
The width available between the seat stays at the bottom of the fender bridge is 2-5/8". Or alternatively measured from center to center on the seatstay tubing at the bottom of the bridge it is 2-15/16".
EDIT 2: The exact spot you should find the serial number on a late 50s early 60s Schwinn if it were there. 61 pictured. The dropout on the left as you are sitting on the bike. But, have you also looked under the bottom bracket? That would imply a much older frame but I would still look.