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

I would use the is_paged() conditional inside index.php to load two separate templates containing your layouts. Something like this:

if ( is_paged() ):
   get_template_part( 'content', 'first-page' );
else:
   get_template_part( 'content', 'paged' );
endif;

Assuming you have two templates, content-first-page.php and content-paged.php.

Edit: If you just want a different template for part of your index page, try this:

if ( !is_paged() ):
   get_template_part( 'content', 'middle' );    
endif;

Put that wherever you want the extra template to load.