That's great. Here's the outline for code that does exactly that. It passes a postponing object to overload::constant, postpones any concatenation and then runs the actual conversion function on stringification at the end.

package My::Package;
use strict;
use warnings;

sub convert {
    YOUR CODE GOES HERE
}

use overload( '.' => \ &_concat,
              '""' => \ &_finalize );

sub import { overload::constant qr => \ &_postpone; }
sub _postpone {
    my $re = shift;
    bless \ $re, __PACKAGE__;
}
sub _concat {
    my ( $a, $b, $inverted ) = @_;
    ($a,$b)=($b,$a) if $inverted;
    $a = $$a if ref( $a ) eq __PACKAGE__;
    $b = $$b if ref( $b ) eq __PACKAGE__;
    
    my $re = "$a$b";
    bless \ $re, __PACKAGE__;
}
sub _finalize {
    @_ = ${shift()};
    &convert;
}


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