Steed's Bowler Hat
The bowler hat is definitely John Steed's trademark: Steed's wearing different bowler hats in the coloured seasons: black, dark blue, dark grey, grey, brown, light brown and white (bleached) ones.

Steed's bowlers are made by Benson and Sons Ltd, now Arthur Benson (The £50.000 Breakfast), he also bought them from Hemming's and Paul in St. James (The Gilded Cage) for 10 Guineas the hat and from Herbert Johnson & Sons (brown bowler in series 5), London and from Barnaby and Sons ( Piers Johnson identified the milliner's mark for series 4 bowlers and TNA episode Hostage).
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Bowler from Herbert Johnson's & Sons Bowler from Barnarby & Sons
He's also wearing a too large or too small bowler hat three times. The hat is oversized in A Change of Bait and Brief for Murder  and he's testing a way too small one in Death at Bargain Prices.
The bowler gets dirty in The Cybernauts.
Katie wearing Steed's bowler when they return to his flat in Lobster Quadrille.
Emma Peel is wearing a (derby) bowler hat twice in Death at Bargain Prices and Silent Dust, but it isn't one of Steed's. There are on set photos or promotional photos with Honor Blackman, Linda Thorson and Joanna Lumley wearing a bowler.
John Steed and his superior from series one and two, One-Ten, with bowler hats in Warlock and Immortal Clay.
Invisible Bowler in Get-A-Way! and Bowler beauty in The New Avengers.
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John Steed's Fashion
Trivia

Several original John Steed bowlers were sold in auctions from 1995 on.
Fashion Props