[Qhttp://m.ebay.co.uk/itm/291730202263UOTE=Carlycu5tard;1840914]http://www.ebay.co.uk/itm/EARLY-WORLD-WAR-2-MK2-BRITISH-TOMMY-ARMY-HELMET-1937-38-/391421116053[/QUOTE] http://m.ebay.co.uk/itm/291730202263
http://m.ebay.co.uk/itm/291730202263[/QUOTE] https://www.amazon.co.uk/The-Discovery-Store-Beer-Hard/dp/B006TYRB3A
You could fit a polystyrene insert in your flat cap, or put an oversize flat cap over a cycling helmet maybe?