How can I add folders to my template directory and have Twig find them?
In your "task" module you must have a task_theme function where you've defined your templates. In there you'll need to update with your new paths too. e.g. 

This

'template' =&gt; "field--expire-date.html.twig"


becomes:

'template' =&gt; "fields/field--expire-date.html.twig"