What item of clothing did Holden buy during the fencing team’s trip to New York in The Catcher in the Rye?

I have to admit that bullgatortail has answered this
question very well, but I am happy to provide a few more details for you.  The simple
answer to your question is that Holden bought a hunting hat that was both red and had
"long peaks" during the famous Fencing Team's trip to New York
City.

I suppose we could equate this purchase to Holden's
desire for things (and people) who aren't "phony."  This hat certainly isn't phony and
is quite utilitarian (at least, for a hunter it would be).  The hat is described as
follows:

readability="5">

[Holden bought] a red hunting hat with very long
peaks.

Holden is especially
proud that the hat only cost him one dollar.  Further, and again in an attempt not to
appear "phony," Holden appears "corny" while wearing the hat backwards in order to
flatter his appearance even more.  Meanwhile, Holden tries to redeem himself a bit after
losing all of the equipment on the subway.  (The poor Pencey Fencing Team!)  I suppose
we can consider that Holden now has only his good looks, his wit, his attempts at being
legitimate (and not "phony"), ... and his red hunting hat.