Ahern played with continuity in "The Squirrel Cage" from the early years. We've seen some examples from 1937 with the annoying neighbor. In 1938 there was a fairly long sleepwalking sequence (two examples below).
However, this was not really continuity of story, but of theme. The first real continuing story that I've found comes in 1939, when Merlin the Magician is introduced (who will, of course, help our friends get rid of that "annoying pest"). However, The Little Hitchhiker gets hold of Merlin's wand. This story lasts for several months, with the magician always failing to get the wand. Here are some examples, including the strip where Merlin loses the wand:
I don't have the finale, so I'm not sure how this story ends, but by 1940 we're back to more stand-alone strips. Ahern experiments with some word puzzle / pun strips, which are clever, but not really that funny. Maybe someone tells him they're not so great, or maybe he just realizes it himself, and he returns to the old formula. But by 1942 he would introduce a new and very different character that would lead the strip into fantastic adventures...!