Towards the beginning of J. D. Salinger's The Catcher in the Rye, Holden buys himself a new red hunting hat:
I took off my coat and my tie and unbuttoned my shirt collar; and then I put on this hat that I'd bought in New York that morning. It was this red hunting hat, with one of those very, very long peaks. I saw it in the window of this sports store when we got out of the subway, just after I noticed I'd lost all the goddam foils. It only cost me a buck. The way I wore it, I swung the old peak way around to the back--very corny, I'll admit, but I liked it that way. I looked good in it that way.
The Catcher in the Rye, chapter 3
This hat then regularly pops up throughout the story, in various places, usually preceding important events in the story. For instance, it's mentioned before Holden and Stradlater have a fistfight:
"Where'dja get that hat?" Stradlater said. He meant my hunting hat. He'd never seen it before.
I was out of breath anyway, so I quit horsing around. I took off my hat and looked at it for about the ninetieth time. "I got it in New York this morning. For a buck. Ya like it?"
Stradlater nodded. "Sharp," he said. He was only flattering me, though, because right away he said, "Listen. Are ya gonna write that composition for me? I have to know."
And afterwards, when Holden is a mess:
I kept sitting there on the floor till I heard old Stradlater close the door and go down the corridor to the can. Then I got up. I couldn't find my goddam hunting hat anywhere. Finally I found it. It was under the bed. I put it on, and turned the old peak around to the back, the way I liked it, and then I went over and took a look at my stupid face in the mirror.
It's mentioned when Holden leaves Pencey:
When I was all set to go, when I had my bags and all, I stood for a while next to the stairs and took a last look down the goddam corridor. I was sort of crying. I don't know why. I put my red hunting hat on, and turned the peak around to the back, the way I liked it, and then I yelled at the top of my goddam voice, "Sleep tight, ya morons!" I'll bet I woke up every bastard on the whole floor. Then I got the hell out. Some stupid guy had thrown peanut shells all over the stairs, and I damn near broke my crazy neck.
Upon arrival at the Edmont Hotel, where Holden later has the encounter with the prostitute and the elevator boy:
We got to the Edmont Hotel, and I checked in. I'd put on my red hunting cap when I was in the cab, just for the hell of it, but I took it off before I checked in. I didn't want to look like a screwball or something. Which is really ironic. I didn't know then that the goddam hotel was full of perverts and morons. Screwballs all over the place.
When thinking about Phoebe:
I took my old hunting hat out of my pocket while I walked, and put it on. I knew I wouldn't meet anybody that knew me, and it was pretty damp out. I kept walking and walking, and I kept thinking about old Phoebe going to that museum on Saturdays the way I used to. I thought how she'd see the same stuff I used to see, and how she'd be different every time she saw it.
Leaving his parents' place after talking to Phoebe, he then gifts the hat to her:
Then I finished buttoning my coat and all. I told her I'd keep in touch with her. She told me I could sleep with her if I wanted to, but I said no, that I'd better beat it, that Mr. Antolini was waiting for me and all. Then I took my hunting hat out of my coat pocket and gave it to her. She likes those kind of crazy hats. She didn't want to take it, but I made her. I'll bet she slept with it on. She really likes those kind of hats. Then I told her again I'd give her a buzz if I got a chance, and then I left.
It gets taken off and chucked at him when he tells Phoebe she can't run away with him, and then is given back later:
"Didja have your lunch? Ya had your lunch yet?" I asked her.
She wouldn't answer me. All she did was, she took off my red hunting hat—the one I gave her—and practically chucked it right in my face. Then she turned her back on me again. It nearly killed me, but I didn't say anything. I just picked it up and stuck it in my coat pocket.
[...] Then all of a sudden she gave me a kiss. Then she held her hand out, and said, "It's raining. It's starting to rain."
Then what she did—it damn near killed me—she reached in my coat pocket and took out my red hunting hat and put it on my head.
"Don't you want it?" I said.
"You can wear it a while."
The hat's frequently mentioned surrounding significant events surrounding Holden in the story, athough some of the mentions not quoted here don't seem to quite fit that pattern.
What is the significance of the hat? What role does it play in the narrative of The Catcher in the Rye?