Changing your SMTP server configuration that way would break things, so the question is whether to set up a new, company-internal SMTP server, and give your employees new addresses there. But that won't quickly stop the phishing, because your ppl still need to get email via the public network from clients and suppliers.

Setting up a new server isn't easy unless you hire an outside service provider, and if you're willing to do that, Slack et al offer a nicer UX than the well known email/webmail clients.

Orgs with sufficient IT resources commonly do run internal SMTP servers.

I meant that as a suggestion compared to designing a new protocol.

Yes I'm old enough to remember when organizations had email but it was internal-only. Probably less for security reasons at the time than that they simply didn't have an internet provider. There were also mainframe-based email systems that were internal to that network.