I am talking about a WHM/Cpanel setup. Is there a way to convince the Exim mailserver (MTA) to get any of the hosted domain name in the "HELO" greeting when dispatching mail? Many ISPs deny accepting mail when the smtp domain name in the setting of a mail client program does not match the vps main hostname.

The only workaround seems to be to change the smtp server name for all domains to be identical to the main server (setup) name.

In detail: When you setup more then one domain on your vps and use your own dns/bind, then CPanel installs the domain records in a way that mail exchange is set to e.g.
firstdomain.com. 14400 IN MX 10 mail.firstdomain.com.
...
mail 14440 IN A 111.111.111.111


which basically instruct your vps that email client programs can connect to the "firstdomain.com" host for pop (imap) as well as smtp using the subdomain name mail.firstdomain.com with the respective ports.

Now, Cpanels MTA (mail transfer agent) exim uses only the vps setup domain name as mailserver address and all Outgoing eMails always show the setup domain or one of its subdomains (depending on setup) in the mail header, no matter which of your vps domains sends the mail.

This way you most likely run into the situation that a couple of ISPs deny accepting your (or your clients) eMails because your (actually one and only) "mailserver does not have their host name in the greeting".

technically, if you telnet (no need to ssh) to mail.firstdomain.com on port 25 the response will be
220-setup.maindomain.com ESMTP Exim 4.52 #1 Sun, 26 Mar 2006 ...
220-We do not authorize the use of this system to transport unsolicited,
220 and/or bulk e-mail.
and, in case you send a helo command to this connection, the result is
helo
250 setup.maindomain.com Hello [11.22.33.44]

That means some mails sent out with mail.firstdomain.com might be rejected as spam.

Is there a way to get exim to display
250 mail.firstdomain.com Hello [11.22.33.44]
or
250 mail.seconddomain.com Hello [11.22.33.44]
etc

I konow, it's a sophisticated question. Anyone who could drop in some thoughts on that ?