There are a million ways a boy in London at that time could get a hat. Like stealing. Which Bae was clearly doing, as we see him break into the Darling household. Also, that hat is fairly typical of boys in that period. In movies like Oliver Twist and the like, boys almost always wear those hats. It’s a signifer of time and location and status. That hat says: poor, orphan, street urchin. Here. All I did was Google “Oliver Twist movie” and these are the images that came up.
We already know that A and E were making Oliver Twist references with Bae in SSTR, the hat just adds to that subtext. It’s not a conspiracy or a clue. It is a way the costume designer locates the character, and more importantly, the audience in time and place.