About the hat...
I believe there are two particularly strong reasons to photograph a person wearing a hat such as this one: either because it is completely out of the person's character to wear it or because it is completely within the person's character to wear it. We already know from Kim that the latter situation applies in this case.
The hat adds a nice symmetry to the image by creating a shadow underneath the hat that is comparable to the shadow underneath his chin. That shadow also frames his eyes in a way that makes me think his eyes are concentrating on something; lacking that shadow, I doubt that I would feel that way.