[jefelump]

Alito's reference to hunting is very minimal in the entire document. He was simply offering the argument, and then debunking it.

Here's the entire paragraph, to give better context:

JUSTICE STEVENS points to a study by amici supposedly
showing that the phrase “bear arms” was most frequently
used in the military context. See post, at 12–13, n. 9;
Linguists’ Brief 24. Of course, as we have said, the fact
that the phrase was commonly used in a particular context
does not show that it is limited to that context, and, in any
event, we have given many sources where the phrase was
used in nonmilitary contexts. Moreover, the study’s collection
appears to include (who knows how many times) the
idiomatic phrase “bear arms against,” which is irrelevant.
The amici also dismiss examples such as “ ‘bear arms . . .
for the purpose of killing game’ ” because those uses are
“expressly qualified.” Linguists’ Brief 24. (JUSTICE
STEVENS uses the same excuse for dismissing the state
constitutional provisions analogous to the Second Amendment
that identify private-use purposes for which the
individual right can be asserted. See post, at 12.) That
analysis is faulty. A purposive qualifying phrase that
contradicts the word or phrase it modifies is unknown this
side of the looking glass (except, apparently, in some
courses on Linguistics). If “bear arms” means, as we
think, simply the carrying of arms, a modifier can limit
the purpose of the carriage (“for the purpose of selfdefense”
or “to make war against the King”). But if “bear
arms” means, as the petitioners and the dissent think, the
carrying of arms only for military purposes, one simply
cannot add “for the purpose of killing game.” The right “to
carry arms in the militia for the purpose of killing game”
is worthy of the mad hatter. Thus, these purposive qualifying
phrases positively establish that “to bear arms” is
not limited to military use.