How to make a content type field unique per user
I don't know the Unique Field module, but I have a suggestion instead.

Create a little utility module which alters the node form of that node type and adds a custom validation. That validation could check for anything on the form, date inclusive.

In the Drupal 6 days it worked like this. I hope this is enough for a starting point. [Also, reading the book Pro Drupal 7 Development can help you a lot if you haven't written a module yet.]