As you already know, Office 365 provides a service dashboard helping you to be aware of any service issue.
That said, this service dashboard may be unavailable when the issue is affecting the authentication process (either authentication itself or MFA services).
Well, you can now have email notification when some services are degraded or are experiencing issue.
To configure the email notification, logon to your Office 365 administration portal (https://admin.microsoft.com/) and reach out the Health\Service Health blade
There, you will find an Edit preference option, shown just below the tabs
Enable the email notification and provide up to 2 email addresses as well as for which Office 365 services you want to be notified
I would recommend you add at least one of your IT support DL/Office 365 Groups or Teams