What hook can I use to modify custom post data before it is displayed on the page?

You can use the_content filter to modify content before it’s output.

function my_the_content_filter( $content ) {
    // maybe limit to archive or single cpt display?
    if( is_post_type_archive( 'your-cpt' )
        || is_singular( 'your-cpt' ) ){
        global $post;
        $meta = get_post_meta( $post->ID, 'some-key', true );
        if( false === $meta ){
            return $content . 'some extra content';
        }
    }
    return $content;
}
add_filter( 'the_content', 'my_the_content_filter' );