@Reyn,
the walking stick part of the umbrella was made of some kind of plastic - perhaps bakelite - and was collapsible . so when it was removed one could easily collapse it- probably no longer than 20 cm total and put it into a briefcase or even just into a pocket .
the dry umbrella was folded tightly and would slip easily into the shell . of course , when dad wasn't at home , we'd play with it ! <GRIN>
of course , you know this one .
on holland america line cruiseships , for "dutch dinner" night , the ladies all receive this kind of hat and the men are given a "dutch boy" cap - have to look for a pix .