It seems that when I go to https://mail.company.com/owa from within the organization, it launches OWA (Exchange Server 2013) just fine, and I'm able to authenticate through to get my mailbox.  However, when I am about the organization, and I go to the same OWA URL, I get the webmail login page for the old mail server (SBS 2008/Exchange Server 2007).  Can anyone assist me with this?  Thank you.

Do you mean you access from Internet by "about the organization"?

The problem is at the public DNS server of company.com. The A (or CNAME) record of mail.company.com is resolved to the old mail server.