You can add more tabs by creating a custom template for product page. For example after or before the tab container in product-description.php or in product-attributes.php.
When you add the following HTML it will become a tab:
</p> <div id="extra_product_tab" class="product-tab"> <p> </p> <h3 class="catalog-header">Your Tab Title</h3> <p>Your tab content. </div>
Custom Product Page Tab without direct access to FTP
With Product Page Customizer active you can achieve it without the direct access to your server files and also have a unique content for each tab for each product. You can go to Catalog > Settings > Catalog Design > Single Page Design and find the Product Page Tabs section.
You can add unlimited number of tabs by clicking the Add Tab button. Each tab should have unique name set in the name field.
After you define your custom tab names you will see a separate editor box for each tab on product edit/add screen.
Custom tab content for each product
With Product Page Customizer active an additional field is automatically created when you add a tab. You will be able to define custom content for each product with the same editor as in description and short description.