You can hook the login_form action:
/**
* Fires following the 'Password' field in the login form.
*
* @since 2.1.0
*/
do_action( 'login_form' );
which is called between the password box and the ‘remember me?’ checkbox, and write out your hidden input field there, e.g.
function wpse_366921_hidden_login_field() {
?>
<input type="hidden" name="hidden-login-field" value="1" />
<?php
}
add_action( 'login_form', 'wpse_366921_hidden_login_field', 10, 0 );