Also, Melville has a thing for hats. In Le Samourai Delon's character is defined by his fedora, which is almost literally his samurai helmet. There's a scene toward the end where Delon puts on his hat and checks in the mirror to be sure the brim is at the right angle before going out to ah, the rest would be a spoiler.

I just thought that scene was to show how the Germans had infiltrated everything in France at the time. Even the local bar and pick up place for women was in the hands of the Nazis.

In addition to Felix and his bowler, hats are significant in several other scenes. The scene where Felix encounters Jean-Francois at the bar begins with a shot of just the German officers' caps on racks (I think it was the NY Times critic who likened these to weapons). When Gerbier is leaving his room in London, he makes a point of removing his hat and leaving it on the bed before he goes. The next shot is of the Gestapo officer entering the room where Felix is handcuffed; the officer's right-to-left movement across the room continues Gerbier's from the previous scene, and ends with the officer removing his hat and placing it on his desk.
There might be other instances, but those are the ones that I remember off the top of my head. There might also be something to the beret that Dounat is wearing until he is killed.
What does it all mean? I don't know. The NY Times critic thought this: "It is a world in which a man's hat is an emblem of his professionalism, part of the armor he dons for battle. When a Resistance member walks into a bote in "Army of Shadows," and Melville shows us a row of Nazi caps neatly lined on a shelf, it's as if he were showing us a cache of weapons."

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Interesting, seldom see anyone in this film carrying walking sticks though.

I think the French police (detectives) wore bowler hats at that time as part of their uniform (see also the Thompson Twins in "Tintin").
Felix was posing as a policeman in that scene. The Vichy police were de facto collaborators with the Germans, Felix didn't like dressing as an agent of the Vichy regime.
I used to want to change the world. Now I just want to leave the room with a little dignity.

It was a symbol of bourgeois conventionalism. The bowler is a disguise: it makes Felix look like a solid French citizen of the sort who was quite comfortable collaborating with Vichy and the Germans. Felix would probably have preferred to wear the working man's beret.

I thought Felix had his poison pill inside the bowler that he lost when arrested . Therefore he couldn't commit suicide.

There's a scene between these two (with Felix) I think. When he's on the train, he takes off his hat and drops his arm to his side. Then we see that neither hand has a hat (when he meets Jean-Francois in the cafe), implying that he dropped it on the train. Later however, he is wearing another hat that falls off when he's arrested in the alley.

You're thinking of Gerbier, who didn't wear a bowler.
Early in the film, Felix wants to take his bowler off, but Gerbier tells him that he needs to wear it as long as he's in the Resistance. Later in the film, when Felix is arrested by the Germans, his hat falls off into the alley, and as the Germans drive off with Felix, the camera lingers on his hat lying in the alley.
I've no answer to the original question, so hopefully somebody can fill us in!

How could he wear his bowler hat while he was parachuting back into France?