Organization of work with El.mail

IMAP will help you

Redirect all emails to your new account email to gmail for example.
It "Redirect" and not forward.

In Gmail in the web interface visible to the sent message under condition that the were sent via SMTP Gmail.

Gmail also has offline...

IMAP of course can help, but be aware that it is configuring your email client.
If in ThunderBird (Yes, I'm sure everyone has this setting) to set a local folder for sent messages, the server does not appear.
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Hence the conclusion — set up your mail client to use IMAP and make sure that it is configured to save sent mail on the server in the desired directory.
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The second option (I often use it, as it works on POP3 Protocol) to configure the client to automatically add your address in "Bcc". Then you will receive a copy of their correspondence in the INBOX. Well, and then decide what to do with it — either themselves manage (delete/copy), or set up automatic rules for such copies (copy to folder).

If the email from the hosting provider doesn't support IMAP You need to migrate from hosted email to Google Apps — https://www.google.com/a/cpanel/premier/new?hl=ru
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Then send all emails with The Bat in the gmail Inbox, sort by Sender in there and throw the mail in the Sent
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Or you can send mail from The Bat to Outlook and use the utility to migrate from Google — tools.google.com/dlpage/outlookmigration