Clear it before calling walk.

foreach my $r (@array){
    my @big;
    walk($r);
    ...
}

sub walk {
    my $i = shift;
    push @big, $i;
    for (@{$hash{$i}}){
        walk($_)
    }
}
[download]

However, it's bad practice for a sub to modify some private variable somewhere. Fix:

foreach my $r (@array){
    my @big;
    walk(\@big, $r);
    ...
}

sub walk {
    my $big = shift;
    my $i   = shift;
    push @$big, $i;
    for (@{$hash{$i}}){
        walk($big, $_)
    }
}
[download]

or

foreach my $r (@array){
    my @big = walk($r);
    ...
}

sub walk {
    my $i = shift;
    my @big = $i;
    for (@{$hash{$i}}){
        push @big, walk($_);
    }
    return @big;
}
[download]