What is the significance of the scene when Holden gives Phoebe his hunting hat in The Catcher in the Rye?

This event occurs at the end of Chapter 23, and comes as
Holden is leaving his parents' appartment and Phoebe to spend the night at Mr.
Antolini's place. Note how the text seems to suggest that Holden decided to give Phoebe
his hunting hat in a spontanous act:

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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.

However, we need to be
aware of the wider symbolic significance of Holden's red hunting hat. It is a bright,
distinct and unique colour, and seems to operate in the novel as a symbol of Holden's
desire to be unique and separate from everybody else. Thus, the way that, at this stage
in the narrative, that he insists on giving it away to his sister, indicates a triumph
in the internal conflict we see raging within him for the need Holden feels for
companionship vs. his need for isolation. He has just spent time with the one character
who he meets that he really connects with in every sense, and thus his willing sacrifice
of his hunting hat is a symbol of his recognition that he does have ties and relations
are important to him.