My answer is a little tangential, but the question reminded me of a story by James Thurber titled "One is a Wanderer." It was reprinted in the collection titled The Middle-Aged Man on the Flying Trapeze and again in The Thurber Carnival. The mood of Thurber''s story is very similar to that of The Catcher in the Rye although there are many differences. Salinger's novel is about a teenager, while Thurber's story is about a middle-aged man. But both are lonely wanderers looking for something in New York, a city where that "something" is notoriously in short supply. They are looking for human contact, sincerity, friendship, even love that doesn't have to be bought and paid for. The protagonist of "One is a Wanderer" is not given a name. It seems fair to assume that Thurber is writing about his own personal experience. Like Holden Caulfield, he knows his way around Manhattan. And like Holden Caulfield, he is constantly on the move, thinking he might go here or he might go there. For example:
I think maybe I'll call the Bradleys, he thought, getting up out of his chair. And don't, he said to himself, standing still a moment, don't tell me you're not cockeyed now, because you are cockeyed now, just as you said you wouldn't be when you got up this morning and had orange juice and coffee and determined to get some work done, a whole lot of work done; just as you said you wouldn't be but you knew you would be, all right. You knew you would be, all right.
Like Holden who keeps thinking about Jane Gallagher, Thurber's protagonist has a woman in his life who never appears in the story. Her name is Marianne. He thinks about her but is reluctant to go to see her--perhaps because he is a little afraid she might not want to see him. Instead he considers seeing everybody else who might be available.
There were several people in Dick and Joe's that he knew. There were Dick and Joe, for two--or, rather, for one, because he always thought of them as one; he could never tell them apart. There were Bill Vardon and Mary Wells. Bill Vardon and Mary Wells were a little drunk, and gay. He didn't know them very well, but he could sit down with them. . . .
There is such an impressive similarity between James Thurber's "One is a Wanderer" and J. D. Salinger's The Catcher in the Rye that it seems likely Salinger was inspired, consciously or unconsciously, by reading the older author's melancholy, introspective tale. --But what about the original question, "What is a good choice of a song that would describe The Catcher in the Rye?" In "One is a Wanderer" the protagonist keeps remembering a jazz song that was very popular in his day.
He became conscious of the song he was whistling . . . and walked out of the room to the elevator, and there he began to sing the last part of the song. "Make my bed and light the light, for I'll be home late tonight, blackbird, bye bye."
So this song titled "Bye Bye, Blackbird," though it dates back to the era chronicled in F. Scott Fitzgerald's novel The Great Gatsby, might be appropriate as a choice for a song that would describe The Catcher in the Rye. The song, for those who remember the sad and sprightly tune, haunts Thurber's story, even when the lyrics are not being quoted. He repeats some of the refrain at the very end of "One is a Wanderer."
When he got to his room, he lay down on the bed a while and smoked a cigarette. He found himself feeling drowsy and he got up. He began to take his clothes off, feeling drowsily contentented, mistily contented. He began to sing, not loudly, because the man in 711 would complain. The man in 711 was a gray-haired man, living alone . . . and analyzer . . . a rememberer . . .
"Make my bed and light the light, for I'll be home late tonight . . . "
This is obviously a great question that could be answered in a number of different ways, depending on the kind of themes that are apparent in your own reading of this text. For me, one of the key and tragic themes is that of loneliness. The way in which Holden chooses to isolate himself from everybody else around him is very moving and sad. Note, for example, how even in the first chapter, he has clearly chosen to keep himself away from everybody else:
Anyway, it was the Saturday of the football game... I remember around three o'clock that afternoon I was standing way the hell up on top of Thomsen Hill... You could see the whole field from there, and you could see the two teams bashing each other all over the place... You could hear them all yelling.
This is our first indication in the novel that Holden is a character who deliberately seeks to keep himself away from everybody else, even though ironically he actually desires human company. Think of when he goes into a phonebox and stays there for twenty minutes, wanting to ring somebody but not being able to select anybody. An excellent song that deals with the theme of loneliness is "Eleanor Rigby" by the Beatles. This would be a good song to select in order to highlight this particular element of the text, as just like Eleanor Rigby and Father Mackenzie in the song, Holden is a character who is desperately lonely and craving relationship with humans.