I would like to catch the outgoing emails to add some custom fields before sending them.
There is only a “notify” method in the Cloud rest API doc, but I guess it is only to add new emails and not for edit the existing ones. Am I right?
Is there any way to reach the outgoing queue?
Not exactly. In order to modify standard email notifications from JIRA cloud you will need to use a cloud based addon such as JEMHC or write your own that implements everything required (Webhook capture, templates, outbound mail config and more).