We have an eircom.net email address and have moved providers several times, I have had UPC, BT and now use Eircom again.
We kept our @eircom.net addresses all the time. And Outlook works fine, we don't need to use the webmail only.
That's a problem with your settings you have.
As someone said you need to look to see have you the correct STMP outgoing and incoming server names in your Outlook setup.
If you change providers you just get the providers' STMP settings to put into your Outlook setups. There's usually a guide on their internet support page.