Change POP account to IMAP

Recently my ISP (iinet) decided to not support email any more. It has an agreement with The Messaging Co (TMC) to move users across. I've setup a new webmail accounts at TMC and am trying to get my Mail accounts changed over. The mail addresses at TMC are the same as they were at iinet, i.e. username@iinet…

iinet are still supporting email for the time being, so the old iinet mail server settings still work. I have managed to transfer iinet IMAP accounts across to TMC and they're working (send and receive mail).

However one of my iinet accounts is POP, TMC only supports IMAP. I tried simply putting in IMAP and SMTP settings for TMC into the Mail account settings (per details by TMC and used for the other accounts), however when I select save a dialog appears after a while that says "Unable to verify account name or password." So I close the settings dialog and don't save changes.

Strangely, it still saves the details for the outgoing SMTP server, but the incoming POP details stay the same and the account still gets incoming email. But I can't send email from the account.

I saw advice here that I could create a new IMAP account in Mail. I tried that, but because the email address is the same (user@iinet…) at step 1 I'm told the account already exists so can't go any further.

Short of deleting the POP account and creating a new IMAP account (which I suspect will delete all emails in the POP account), how do I change Mail settings from POP to IMAP?

PS. I tagged the question "Other Apple applications and sevices" because Mail isn't listed in the Apps and services topic category (or anywhere else that I could find).