The hunting hat must symbolize an idea or concept.
It never goes anywhere. Morrow on the train to New York. If they should run out of the field, the momentum would send them running over the cliff.
Holden responds to this vulnerability by taking her in his arms and kissing her in a pure and loving fashion. His position is a metaphor for his views on life. Affectionate and emotional They got drunk together once in chapel.
This is a person who has a special place in Holden's life. There are things in the story that that parents may not have known, however. Jane was vulnerable. Salinger has made you an integral part of the novel. Good player but a big phony Holden and Sunny only talk.
The hat may remind Holden of someone else, but that someone else must be connected to that greater idea or concept that is important to Holden and, of course, important to the theme.