iOS 3rd-party mail app which allows connections to multiple ports at the same time I need to travel to a country that blocks port 25 (SMTP) and 993 (IMAP) country-wide. 

The stock mail.app does not allow me to specify multiple ports for both SMTP and IMAP. 

Changing the port values back and forth for up to 10 email accounts is very tedious…

Does anybody know if there's a third party mail app client that allows me to specify, say, "25, 1225" and "993,9993"?
I do not know of a third party application. But besides a vpn, you could try going to settings>mail>your specific mail account. This may give you the option to change the incoming mail server, but it may not. But you can change the outgoing mail server by adding a entire new secondary server in case the first one doesn't work. You can look at the pictures attached.