I think it's pretty ironic that you brought those up as examples...I'll explain why.
The first problem I had with his hat was identifying what kind of hat it was. My initial guess was an officer's cap (which is what you showed, and it makes sense...I mean, he thinks he's a detective after all), but after discussing it amongst various people, we actually came to the conclusion that it most resembled a newsboy cap (or more specifically, a spitfire cap). Examples of these are as follow:
-http://www.kangolstore.com/is-bin/INTERSHOP.enfinity/WFS/Hats-Kangol-Site/en_US/-/USD/ViewProductDetail-Start;pgid=_FWgI0oUaoI000w69zKCHyZq00002hmYaFzO?Pr oductUUID=nYbAqAp4e8UAAAEWcaqCz0Q4&SGCatalogCatego ryID=RybAqAqvZy0AAAEGWwKqFweG&JumpTo=OfferList
I actually think the 3rd one is the closest, but of course...they don't have the color I'm looking for.
Now what's ironic about this all is the the Japanese cosplayers I've seen do Naoto (which I'm sure you've seen too) also picked what appears to be officer's caps...
Regardless, I really like the first example you showed me, and it has also become a candidate, thanks!