How to redirect on node save to fill out extra fields attached to terms
I've found there's a new module called Entity Connect which adds a button to entity reference fields for creating new references on the fly.  The module sends the user to the entity creation page (add term, etc) allowing you to fill out all necessary fields, then redirects back to the original node form.

This works great.  Now the only thing necessary now for my use case is to figure out how to programmatically add taxonomy terms based on the value(s) of the entity reference field.