I'm sorry, but a sports jacket, blazer or suit worn with an open-necked shirt looks bad to me - sloppy and unfinished. I am aware that this look was sometimes sported in the "Golden Era", but that period is not my ideal; I look more to the period around 1960, when such a look was considered bad form - it had disappeared by about 1955 and did not reappear until the early 1970s. Now (sadly) it's everywhere.
I don't know about France, but in Britain hat-wearing was on the decline by the mid 1930s.I am also surprised how many men are sans hat or cap.