So if you have 100 clients you have to set 100 names in there. That is not the normal way to work and wrong solution to be used.
Normally all clients are allowed to use SMTP connections anyway, so you most likely have the option "use this server to handle users mail" or something still selected on the MX page of the users. Or some other configuration issue.
Anyway, it's loads of work to specificy all users. I would suggest to set it back to the default setting and then just...
if only for one user, you can allowed some user for external smtp connection.