No content found on page 2 of pagination with 1 post per page

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I don’t know what largo_content_nav() does but you clobber the main query here:

query_posts('posts_per_page=1');

And then several lines later attempt to paginate (or so the function name and argument suggest):

largo_content_nav( 'nav-below' );

If largo_content_nav uses the global variable $wp_query which I kind think it must since you don’t pass it a WP_Query object, then you are paginating based on an object that is different from the one used to load the page. That is:

  1. Long before the theme templates load, WordPress creates the
    $wp_query global based on various query parameters and rewrite
    rules.
  2. The page loads according to information in that query
  3. You, by using query_posts which you should never do, then
    overwrite $wp_query
  4. And attempt to paginate. Unfortunately you’ve altered the query at
    step #3. Things are out of sync.

I don’t honestly understand (or I am not sure that I understand) what you are trying to accomplish with this block of code, but your comment below …

All I needed to do was show 1 full post per page on the archives for
this taxonomy term.

… makes me think that all you want to do is limit your archives to a single result per page. You have two loops on the page, but it seems that you only want one and have cooked up something odd and wildly complicated. I think what you need is:

function single_per_page_archive_wpse_133857($qry) {
  if ($qry->is_main_query() && $qry->is_archive()) {
    $qry->set('posts_per_page',1);
  }
}
add_action('pre_get_posts','single_per_page_archive_wpse_133857');

And remove most of this:

            // and finally wind the posts back so we can go through the loop as usual

            rewind_posts();
query_posts('posts_per_page=1'); 
            while ( have_posts() ) : the_post();
                get_template_part( 'content', 'singleseries' );
            endwhile;

Keeping just this (I think):

get_template_part( 'content', 'singleseries' );