You can add Custom Fields for Address in Customize/Address of the Setup section.
Fig 1: Adding Address Custom Field
Read more on types of Custom Fields here. Note: Sections and Page Workflow do not apply to Address fields.
Address Custom Fields automatically appear in Accounts and Contacts.
Fig 2: PO Code and Main Entry Custom Address Fields in Account
If Contacts inherit Address from Accounts, then they will be pre-populated when a new Contact is created from an Account.
Read more on inheriting Address here.