Custom fields disappear if a customer has the address specified in his account. How to make these fields reappear?

Our Checkout's custom fields (which can be managed at System > Amasty One Step Checkout ) are being processed and saved as customer's address.

These fields' content can be edited as a part of existing address adjustment and new address adding.

The Order Attributes for Magento 1 extension can add address-independent custom Checkout fields if needed.