Find <div id=tab2_login" and replace it with following code.Assuming you will change CSS and JS accordingly.
<div id="tab2_login" class="tab_content_login" style="display:none;">
<h3>Register for this site!</h3>
<p>Sign up now for the good stuff.</p>
<form method="post" action="<?php echo site_url('wp-login.php?action=register', 'login_post') ?>" class="wp-user-form">
<div class="username">
<label for="user_login"><?php _e('Username'); ?>: </label>
<input type="text" name="user_login" value="<?php echo esc_attr(stripslashes($user_login)); ?>" size="20" id="user_login" tabindex="101" />
</div>
<div class="password">
<label for="user_email"><?php _e('Your Email'); ?>: </label>
<input type="text" name="user_email" value="<?php echo esc_attr(stripslashes($user_email)); ?>" size="25" id="user_email" tabindex="102" />
</div><!--New Field Here -->
<div class="username">
<label for="user_login"><?php _e('New Field Name'); ?>: </label>
<input type="text" name="new_field" value="<?php echo esc_attr(stripslashes($new_field)); ?>" size="20" id="new_field" tabindex="103" />
</div> <!--End of New Field -->
<div class="login_fields">
<?php do_action('register_form'); ?>
<input type="submit" name="user-submit" value="<?php _e('Sign up!'); ?>" class="user-submit" tabindex="103" />
<?php $register = $_GET['register']; if($register == true) { echo '<p>Check your email for the password!</p>'; } ?>
<input type="hidden" name="redirect_to" value="<?php echo $_SERVER['REQUEST_URI']; ?>?register=true" />
<input type="hidden" name="user-cookie" value="1" />
</div>
</form>
</div>