Sending a testmail from a server with a configured exim is easy as pie.
From your usershell:
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tell exim that we want to send a mail to an recipient and tell us about the details of the sending procedure
$ exim -v receipient@foo.bar
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Tell exim about the sender, the subject and input some text.
From: user@your.domain.example Subject: Foobar Text Text Text I like Pie!
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Press
Ctrl+d
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You should see the details about the SMTP connection. You can return to your shell with
Ctrl+c