For my personal mail server I ended up using the free tier of SMTP2GO as outbound smtp relay. You can just register your domain as a whole, and the setup was pretty painless. They let you send up to 1000 mails/month for free, which is plenty for my own personal use, but if you need any more than that, these relay services can get quite expensive. You usually end up paying the same price as the fully hosted solution that most of those providers offer alongside the relay service.