I’ve a couple of tweed caps from renowned hatters Failsworth, one of which is a baker-boy style, though bought sometime before they were displayed on Peaky Blinders: anyway, it’s brown not de rigueur black.
I take one with me only when it’s raining or rain is expected, including to outdoor sporting events.
What I never cover the barnet with is a bobble hat or its relative the ‘beanie’ both of which I regard as ugly, even if they do keep the ears warm. Better cold than undignified 