I assume the code you added to functions.php is a filter for ‘the_password_form’, where you’ve got something like:
function my_protected_post_form () {
$myCustomizedFormCode="<form action=...."
return $myCustomizedFormCode;
}
add_filter ('the_password_form', 'my_protected_post_form');
So where you have the instructions to the user in that custom form code string, you just need some simple HTML to provide the link you want. For example:
'[...]To view this protected post, enter the password below, or <a href="http://example.com/passwordsignup">request a password</a>.[...]'