Why was Holden trying to “feel” some kind of good-bye?

Holden says that he wants to "feel some kind of good-by." He wants to feel some sort of connection to the place he's leaving, but he doesn't like the school or many of his peers or the staff members. He feels like the school is full of "phonies," especially the adults. In fact, he associates many aspects of the adult world with phoniness and hypocrisy.
Finally, though, he comes up with a memory so he can "get a good-by." He remembers himself, Robert Tichener, and Paul Campbell playing catch in front of the academic building. They threw the ball back and forth until it got dark. Even when they could barely see the ball, they kept playing until a teacher stuck his head out the window and told them to go get ready for dinner.
The scene seems to remind Holden of childhood and the innocence of it. The boys played and played until the sun went down. In that span of time, at least, they had no responsibility, no pressure, no tests. They were just boys playing catch. Then the teacher came to tell them it was time to go get ready for dinner. The boys, in this instance, did not have to fret about time. The adults were worrying about that. The adults would prepare the food and make sure that the boys were where they were supposed to be when they were supposed to be there. The boys had only to be boys, carefree. It is a memory untainted by the things he hates about grown-ups and growing up.
There is nothing that Holden reveres more than the innocence of childhood. The title of the book comes from a mental image he has of children playing in a field of rye and of himself catching the children who get too close to the cliff at the edge of the field, likely a metaphor for saving them from the realities of the world. The phoniness and hypocrisy that he hates in the adults do not infect Phoebe or Allie or the other children in the novel—and he adores them for this.