Update: After further testing, I don't think this is a good solution. The symbol tables do not behave as I thought they would.

Or, for more segregation between the old and new modules, you can copy the entire symbol table.

#!/usr/bin/perl
use strict;
use warnings;

BEGIN {
   package New::MyModule;

   *Old::MyModule:: = \*New::MyModule::;
   %My::Other::Name:: = %New::MyModule::;

   sub foo { print @_ }

   1;
}


New::MyModule::foo("new\n");
Old::MyModule::foo("old\n");
My::Other::Name::foo("other\n");

$New::MyModule::extra1 = 'set as $New::MyModule::extra1';
$Old::MyModule::extra2 = 'set as $Old::MyModule::extra2';
$My::Other::Name::extra3 = 'set as $My::Other::Name::extra3';

# This is to avoid warnings about the above being used only once
my @array = (
        $New::MyModule::extra1,
        $Old::MyModule::extra2,
        $My::Other::Name::extra3,
);

print "\nNew::MyModule:\n";
foreach my $key (keys %New::MyModule::) {
    print "$key: $New::MyModule::{$key}\n";
}

print "\nOld::MyModule:\n";
foreach my $key (keys %Old::MyModule::) {
    print "$key: $Old::MyModule::{$key}\n";
}

print "\nMy::Other::Name:\n";
foreach my $key (keys %My::Other::Name::) {
    print "$key: $My::Other::Name::{$key}\n";
}
[download]

Which produces:

new
old
other

New::MyModule:
extra2: *New::MyModule::extra2
extra1: *New::MyModule::extra1
foo: *New::MyModule::foo

Old::MyModule:
extra2: *New::MyModule::extra2
extra1: *New::MyModule::extra1
foo: *New::MyModule::foo

My::Other::Name:
extra3: *My::Other::Name::extra3
foo: *New::MyModule::foo
[download]