Add date field to drupal entity registration
Goto Strcuture-&gt;Registrations-&gt;YourRegistrationType and add a new field of type date. 

You can use Field Validation Module to make sure that the entered field is between the selected date range of the content type on which you have the registrations enabled.

Note : You could also use the tokens module to make sure that this datefiled in registration type has default value of the starting date of the date field in parent content type.