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 . . . "