Subselect dependent on first select value
You can use 2 subqueries, one joining with the contact_id and another joining with the alt_contact_id. Then UNION them and ORDER BY:

SELECT cont.*, cred.*, 'main' AS contact_type
FROM e_contacts AS cont
  JOIN e_credentials AS cred
    ON cred.contact_id = cont.contact_id
WHERE cont.contact_id = 1

UNION ALL 

SELECT cont.*, cred.*, 'alternate'
FROM e_contacts AS cont
  JOIN e_credentials AS cred
    ON cred.contact_id = cont.alt_contact_id
WHERE cont.contact_id = 1

ORDER BY contact_id ;


If you want this for all contacts, just remove the two WHERE clauses.