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While on either the Edit Contact Form page or the Create New Contact Form page, use the Search Bar to search for an existing field in the Find a field section.
(You do not need to press Enter while searching for a field. Doing so will save the Contact Form and redirect you to the Manage Contact Forms page)
Use the Filters to display fields of specific types
Press the green tick symbol to add the field to the form
Scroll to the bottom of the page and press Save