How can I add my own ‘read more’ or ‘after the jump’ type of link to the article page?

Question

I would customize the Read More tag (natively built into WordPress).

Instead of producing your own URL, this tag will render the permalink for you. If you want to customize the text that appears here you can follow these specific instructions as well.

function modify_read_more_link() {
    return '<a class="more-link" href="' . get_permalink() . '">Your Read More Link Text</a>';
}
add_filter( 'the_content_more_link', 'modify_read_more_link' );