How to send emails on server with 25/465/587 outbound blocked?

If you use port 2525 as @itsbhanusharma recommended it will work with all mail delivery services that I have tested. I have done dozens of installs this way.

Another solution would be to develop a plugin that delivers mail through the API. Thus far, no one has seen the need to do that. If you have a budget of $1000 or more, you could post in Marketplace and see if anyone is interested in developing one.