The Register New Volunteer Screen does not support only the field type "Time".

All types of custom fields from Contact objects can be included in the "Add New Volunteer" screen except the "TIME" field.

The "Add New Volunteer" supports the "Date & Time" field and only the "Date" field. And does not support only "Time" (Which shows only Time) field.

The custom fields which are added in the Contact Fieldset are shown in the "Add New Volunteer" screen.  More information here for adding Contact Custom fields to the "Add New Volunteer" screen.