There are many Transactional Mail Services available. Some of them will work through the SMTP settings, while other also has the option to work through its API.
Remember that some Transactional Mail Services also highlights the importance to insert some settings into your Domain Records.
See some of the lists to work with Transactional Mail
If you want also to customize your email, there are some services that allow you to customize your firstname.lastname@example.org
Some Services may offer both options, while others only work as Transactions per Server Domain without Mailbox.
and also on your WordPress Settings. Go to Settings > General