While trying to learn about Old Put cigars, I found nothing unrelated to cards with the stamp, but.......
I found that the old NY central railroad line from NYC to Putnam, NY was referred to as the "Old Put". It operated from the early 1890s until 1913, I believe (perhaps longer), according to one source.
Given the era is the same, is there a relationship between the Old Put stamp and the railroad perhaps. I'll bet there were more than few cigar smokers riding the Old Put in the day.