Publishing a contact page without showing a corresponding menu item in the main menu
Go to "Menus" -> "Menu Manager" -> "Add New Menu". Call it "Hidden menu", or something similar
Create a Fox Contact Page. Ensure it is published and set "Menu Location" property to "Hidden Menu"
Don't create a "Menu Module" to show your new hidden Menu in your pages, so that this menu remains active, but not displayed anywhere.
How to reach your hidden Fox Contact page
The URL of the hidden contact page may depend on "SEO Settings" in Joomla "Global Configuration".
Use the following URL format if "Search Engine Friendly URLs" option is disabled, but consider that it should always work, regardless of your SEO Settings.
index.php?option=com_foxcontact&view=foxcontact came from menu item's "Details" -> "Link" value
"Itemid" parameter is based on "ID" value
Once in frontend, if you see the message "Please provide a valid URL" probably you forgot to add the "&Itemid=..." parameter in the URL itself.
In this case, and if one or more published contact pages exist, a list of links follows from which you can safely copy the contact page URL.
Other working URLs depending on "SEO Settings" configuration
(Note that in the following examples "contact-us" came from the menu item's "Alias" property).