Different page template for paginated content?

I suppose the obvious alternative is having logic in content-page.php that checks to see if it’s paged content and outputs differently if that’s the case.

Basically.

In content-page.php:

get_header();

//WordPress has a global variable for whether a post/page is paginated or not
global $paged; 

// If post/page isn't paginated
if ($paged === 0) {
    //Do stuff that you want for non-paginated posts/pages
}
//If post/page is paginated
else {
    include 'content-page-paginated.php';
}

get_footer();

(Edit: changed $multipage to $paged as per Nathan’s comment)