In theory, you could bless your code reference $new into the package Foo, in order to associate a type with it:

package Foo;

sub new {
    my $arg = shift;
    my $class = ref($arg) || $arg;  # finally some use for this cargo-
+cult thing :)
    print "creating object of type $class\n";
    return bless {}, $class;
}

package main;
use strict;
use warnings;

my $new = bless \&Foo::new, "Foo";  # code object

# and then one of:

my $obj;
$obj = $new->(ref($new));           # passes "Foo"
$obj = $new->($new);                # passes code object of type Foo
$obj = $new->new();                 # passes code object of type Foo
$obj = $new->$new();                # passes code object of type Foo
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But don't do it, it's too confusing...  Also, you couldn't store/pass around any other args this way, as with the closure.