Show next post in same category but start from the latest

Question

There is no way to do this natively, so you will need to do this by custom functions.

Here is what you will need to do:

(This complete solution is based on the build-in taxonomy, category)

You will need to implement a system whereby you will need to determine the current post position of the current post been viewed in the specific category.
You will need to get the ID of the latest post in the specific category, in other words, the latest published.
Link to the latest post in that category when you are viewing the 7th post and also the link to post 6.
Only show the link to post one on all posts older than the 7th post.
Display both the next_post_link() and previous_post_link() when viewing newer posts than the 7th post

LETS CODE

We need to get the post ID of the single post being viewed, and also the post type if necessary. To get this, you can make use of get_queried_object()

Once we have that, we need to determine to which category the post belongs to. This is done by using get_the_category(). (Take note, this will only work successful when a post belongs to one category only. If there are more than one category, the first one is used).

All this info will be used in a custom query making use of get_posts. As we only need the post ID’s, we are only going to retrieve that.

This returned array of post ID’s is the data that will be used to determine post position and the newest post in that category.

This query can get quite expensive later, so we will save all our data into a transient which I have set to expire every 7 days. All the data needed will be stored in variable called $numbers

We also need to delete and recreate the transients once posts are updated, deleted or published. For this we will use the transition_post_status action hook

With this all set up, we move over to the single.php template. This is where you are going to display what is needed. as stated

Display only newest post link for posts older than 7
Display the newest post link and next_post_link() on post 7
Display both the next_post_link() and previous_post_link() when viewing newer posts than the 7th post

To get the newest post from the category, you will use get_post() with the ID saved in the $numbers variable

This should do it

HERE IS THE COMPLETE CODE

(Modify as needed. I have tested it and it works according to what you stated in your question)

In functions.php

function get_display_post_number() {

    $query_object   = get_queried_object();
    $id             = $query_object->ID;
    $transient_id   = 'post_number_' . $id;

    if ( false === ( $numbers = get_transient( $transient_id ) ) ) {
        $categories = get_the_category( $id );
        $cat        = $categories[0]->cat_ID;

        $post_args = [ 
            'post_type'         => $query_object->post_type,
            'cat'               => $cat,
            'fields'            => 'ids',
            'posts_per_page'    => -1,
            'order'             => 'ASC',

        ];

        $q = get_posts( $post_args );

        $total_posts    = count( $q );
        $post_number    = array_search( $id, $q ) + 1;
        $current_post   = $total_posts - $post_number + 1;
        $newest_post    = $q[$total_posts - 1];

        $numbers = [
            'post_number'   => $post_number, 
            'newest_post'   => $newest_post, 
            'total_posts'   => $total_posts, 
            'current_post'  => $current_post 
        ];

        set_transient( $transient_id, $numbers, 7 * DAY_IN_SECONDS );
    }

    return $numbers;
}

add_action( 'transition_post_status', function ( $new_status, $old_status, $post )
{

        global $wpdb;
        $wpdb->query( "DELETE FROM $wpdb->options WHERE `option_name` LIKE ('_transient%_post_number_%')" );
        $wpdb->query( "DELETE FROM $wpdb->options WHERE `option_name` LIKE ('_transient_timeout%_post_number_%')" );

}, 10, 3 );

In single.php

$a = get_display_post_number();
if( 7 === $a['current_post'] ) { 

    next_post_link('%link', '%title', TRUE);
    echo '</br>';

    $latest_post = get_post( $a['newest_post'] );
    $post_link = get_permalink( $latest_post->ID );

    echo '<a href="' . $post_link . '">' . $latest_post->post_title . '</a></br>';

}elseif( 7 < $a['current_post'] ) { 

    $latest_post = get_post( $a['newest_post'] );
    $post_link = get_permalink( $latest_post->ID );

    echo '<a href="' . $post_link . '">' . $latest_post->post_title . '</a></br>';

}else{

    next_post_link('%link', '%title', TRUE);
    echo '</br>';
    previous_post_link('%link', '%title', TRUE);

}

Here is what you will need to do:

LETS CODE

HERE IS THE COMPLETE CODE

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