Unable to send emails to external domain using SMTP

I am not able to send emails to external domain addresses like 'user.one@asdf.com' using the code below.

SmtpClient smtpClient = new SmtpClient(smtpMailServer);
smtpClient.UseDefaultCredentials = true;
smtpClient.DeliveryMethod = SmtpDeliveryMethod.Network;
smtpClient.EnableSsl = true;

//Sending mail.
smtpClient.Send(mailMessage);

I get an exception -

Mailbox unavailable. The server response was: 5.7.1 Unable to relay for xxx@example.com

If I change the DeliveryMethod to -

smtpClient.DeliveryMethod = SmtpDeliveryMethod.PickupDirectoryFromIis;

I am able to send the emails on my local machine. But it fails on the production site with an exception -

Cannot get IIS pickup directory

Can you please suggest me what to do?