If the old host uses cPanel, you can use the Migration tool in the Control Panel to import your mailboxes from a cPanel backup.
Alternatively, if the other hosting provider can provide an archive of the emails in Maildir format, you can provide us with the file, and we will check if we can import your messages from it.
You can also move the messages via IMAP. You can move messages between servers via IMAP by using a mail program such as Thunderbird. You will need to configure two IMAP accounts for each mailbox in the mail program - one that connects to the old mail server, and one that connects to our server. Then you will be able to drag-and-drop messages between the servers. Note that you would first have to recreate the necessary mailboxes on our end through the Mail Manager in the Control Panel.