Custom /page/2/ template (different from index.php)?

I’m trying to create a custom page 2 template that’s different from the main index.php. For example, the main index page would display the latest news + a lot of features like a magazine. When readers click to read on to page 2, 3, 4 etc of the news posts, they are taken to a more regular news page. Kinda like something on theverge.com (at the bottom, click “next page” and see what I mean).

Anyone have any ideas on how to do this? I know it’s possible to create archive pages for categories and tags, but I’m not sure how to do it for the index.php.

Thanks!