Secondary email address is undeliverable (secondary domain)

I'm probably missing something stupid and simple. But as a 1-man IT department, I don't have a second set of eyes to pick up on my mistake...

A sister company closed its US office and some of the employees moved to our office. Their managed service provider shut down their Exchange server, but assigned ownership of the domain name to us. I duplicated the DNS Zone File information for the second domain name on the GoDaddy web portal to match our primary domain. On my Exchange 2010 server, I added an additional email address to select users such as name@seconddomain.com.

However, when I send a test email from an outside source, I keep getting "Delivery Notification: Delivery has failed" messages:

This report relates to a message you sent with the following header fields:

  Message-id: <ca379176-6b71-482c-92e3-025c360fb2c5@me.com>
  Date: Wed, 28 Oct 2015 14:00:15 +0000 (GMT)
  From: Eric Jack <eajack@blah.com>
  To: ejack@seconddomain.com
  Subject: Test 10:00

Your message cannot be delivered to the following recipients:

  Recipient address: ejack@seconddomain.com
  Reason: Remote SMTP server has rejected address
  Diagnostic code: smtp;550 5.7.1 Unable to relay
  Remote system: dns;mail.mycompany.com (TCP|17.172.109.150|44229|(My IP)|25) (mail.mycomany.com Microsoft ESMTP MAIL Service ready at Wed, 28 Oct 2015 10:00:29 -0400)