You have a ton of bugs and security issues with your version. There is "no alternative" to upgrade! This said, check out the code of Magento 1.9 for the sendPasswordForgottenEmail.
It starts here:
\Mage_Customer_AccountController::forgotPasswordPostAction
generates a password reset token:
$newResetPasswordLinkToken = Mage::helper('customer')->generateResetPasswordLinkToken();
$customer->changeResetPasswordLinkToken($newResetPasswordLinkToken);
$customer->sendPasswordResetConfirmationEmail();
sends an email with this token app/locale/en_US/template/email/account_password_reset_confirmation.html
If you click on it, the token is checked \Mage_Customer_AccountController::resetPasswordAction
, then you can reset your password: app/code/core/Mage/Customer/controllers/AccountController.php:591