Create an email account for your former hosting provider within the new email client using IMAP.
Create a separate account for your new account within the email client, also using IMAP.
Update both accounts to ensure that each account contains all of the current emails.
Select all the emails and folders from IMAP Acct #1 Inbox and drag contents to IMAP Acct #2 Inbox.

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How to migrate an email account with IMAP