If the user adds a new payment method after the subscription has already expired, we won't automatically renew the subscription again. The user needs to purchase the subscription again in order for the billing to start again.
Once a payment method, for example, a credit card linked to an existing subscription expires, the subscription is taken off auto-renew automatically and this is the expected behavior. In this case, the user can subscribe anew to have the subscription on auto-renew again or if you have a grace period enabled, the user is able to update their credit card information in My Account during the grace period.
In order to prevent this situation from happening, you can customize the text in the Credit card expiring notification that can be enabled under Manage → Email Manager, and advise users to add a new card and make sure to delete the old one - the subscription will then become linked to the new card and the user can avoid interruptions to their subscription service.