You have a two nested for loops and so they can only print two levels. The easiest way is to use recursion. The following code has two solutions. get_all_chains will return an array of all of the call chains. print_all_chains will print the call chains as they are found.

#!/usr/bin/perl -w
use warnings;
use strict;
my ($tag, %list);
while (<DATA>){
  next unless /./;
  if (/^(\w+)/){
    $tag = $1;
    next;
  }
  if (/^-- (\w+)/){
    push @{$list{$tag}}, $1;
  }
}

print "get_call_chains\n";
print "---------------\n";
for my $call_chain ( get_call_chains("main") ){
    print join(" ", @$call_chain),"\n";
}
print "\n\n";
print "print_call_chains\n";
print "-----------------\n";
print_call_chains("main");

sub get_call_chains {
    my $root = shift;

    my @calls;
    if( exists $list{$root} ){
        for (@{$list{$root}}){
            push @calls,  get_call_chains($_);
        }
    }
    else {
        @calls = ([]);
    }

    unshift @$_, $root for @calls;

    return @calls
}

sub print_call_chains {
    my @stack = @_;

    my $current = $stack[$#stack];
    if(exists $list{$current})
    {
        for (@{$list{$current}}){
            print_call_chains(@stack, $_);
        }
    }
    else {
        print join(" ", @stack),"\n";
    }
}
__END__

main
-- check
-- check1

check
-- computing
-- net

check1
-- computing2
-- net2

computing
-- community
[download]

which results in

get_call_chains
---------------
main check computing community
main check net
main check1 computing2
main check1 net2


print_call_chains
-----------------
main check computing community
main check net
main check1 computing2
main check1 net2
[download]

get_call_chains can be optimized by storing the result on the first call for a given subroutine, and using the stored value if it is ever called again with the same subroutine.

Note this version does not address the infinite loop if the analyzed code uses recursion.