Hello,
I am not sure I understand your question correctly (please let me know if this does not answer it).
You can show the TE module on a contact page by assigning the module only to the menu item that calls the contact page. If you do not have a menu item for a contact page, create one, assign it to a menu that you will not display and assign the TE module to that menu item.

What I would do in your case is create 2 TE modules, one for the pastoral staff and one for the support staff.
I would associate each module to the menu item that lists the contacts of that category (if no menu item exists, create one for each contact category, publish them and add the menu items to a menu that you will not display). Associate each module to the menu item that is supposed to show the category wanted and you will see a module when showing the category or a contact.

Note that I am looking into creating a plugin for the TE component that will add a 'Go back to index' link to the standard contact page.