Expecting to be ridiculed, another connection I believe may be relevant was the common sight of an ethnic entrepreneur who made his living by playing an accordion, with a monkey who on command would jump onto a stick, which was then pointed towards a potential donor, and the monkey would take off his little hat and hold it out to collect coins for his owner. So, "getting on the stick" was used to mean "start earning my keep." Laugh if you will, but I remember seeing them in Boston many times in the twenties.