Mac Mail suddenly won't connect to outgoing smtp server

Have you tried using Mail Connection Doctor that is built-in macOS? It could help determine the cause when the Mail app is not successful sending messages.

As you can imagine, there are a number of steps that need to be performed to send email.

They include:

1. Doing a DNS lookup to find the IP address of the SMTP server.
2. Initiating contact to that server to find out its capabilities and informing it what your client's capabilities are as well.
3. Establishing a secure link by submitting your user email account credentials.
4. Authentication of those credentials by the SMTP server.
5. ... and finally sending the email message.

Any one of these can fail and would have to be troubleshot to determine which is the culprit.

When the SMTP server isn't working, can you receive emails?

Mail (El Capitan): Mail troubleshooting

Mail (El Capitan): If you can’t send messages

Mail not sending or receiving messages after updating OS X

Are you running any anti-virus programs?

The next time you are having problems, see if it will work in the Safe Mode.

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply.        Pastebin

Etrecheck – System Information

Yes ... but what we don't know is at which of those that if fails when you randomly can't send an email ... right? Without doing some additional troubleshooting, it would be impossible to tell. The problem is that it needs to be done when the Mail client fails to send the message ... and, I'm assuming that it is to only one email provider ... correct?

I am getting a '550 relay not permitted error.' So, that's the problem. But the question remains: "why."

The SMTP "550" error is typical when there is a non-existant email address at the recipient's location. It could also indicate that the recipient's firewall is rejecting the message or that the server is down.

A "551" error would be related to relay issues. In this both your email address and the recipient's are not locally hosted by the SMTP server. This is one method to prevent spamming by the email provider.