I don't really understand what you're doing respectively what your logic is concerning @big. My usual approach to having a recursive subroutine return a list is to either pass around a reference to the result set or to collect the results of the branches:

# Make walk() return a list and collect the results:
sub walk {
    my ($i) = @_;
    my @local_results = $i;
    for (@{$hash{$i}}) {
        push @local_results, walk($_);
    };
    return @local_results;
};

...

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

The alternative is easier if your tree walking somehow has to check for duplicate elements, so it's more an optimization:

# Make walk() augment the result list:
sub walk {
    my ($i,$results) = @_;
    # If we're called with just one argument, we need to return a fres
+h list
    $results ||= [];
    push @$results, $i;
    for (@{$hash{$i}}) {
        walk($_,$results); # here, we don't care about the results of 
+walk()
    };
    return @$results;
};

...

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