How to link to “most recent” document in the Media Library?

You could search the latest attachments like my example below.
I searches for the string “newsletter”, and the first time it pops up in a filename, it writes a link.
I’ve tried to make it easy to understand.

$attachments = get_posts( array(
    'post_type' => 'attachment',
    'posts_per_page' => 20,
    'post_status' => null,
    'post_mime_type' => 'application/pdf,application/msword'
) );

foreach ( $attachments as $attachment ) {
    $url = wp_get_attachment_url( $attachment->ID);
    $needle = "newsletter";
    if(stripos($url, $needle) !== false){
        echo "<a href="https://wordpress.stackexchange.com/questions/133872/$url">";
        echo "Read the latest newsletter here";
        echo "</a>";
        break;
    }
}

posts_per_page sets how many attachments you want to search, so it depends on how much activity the site has got, and how far back it should look for a file matching the credentials.

The only BUT in this is that you should have a specific string represented in every newsletter name uploaded in order for this to work.
So newsletter_feb.doc would work in this example.

Note that you should specify post_mime_type to match the documents your client will be uploading newsletters in. But like this it skips images, which in my experience fills up a lot of space in the media library.