I've successfully run the Hybrid Configuration Wizard in an environment with Exchange 2010 on premises and exchange online.
After it completed I did a remote mailbox move on a test user and found that OWA wouldn't work. Fixed this easily enough by going to EMC, Organization Cinfugration->Organization Relationships, properties, External Organization and adding my email address domains to the "Federated domains of the external Exchange organization". The configuration wizard only had the <domainname>.mail.onmicrosoft.com entry.
After that fix a login to the owa would let the user know the new Exchange Online URL.
I then started Outlook 2013 for the test user, it prompted for credentials, shows trying to connect and then "DISCONNECTED".
Did I miss a step after completing the hybrid configuration wizard? Did it maybe miss another section (like it did for OWA) that I need to manually fix to get Outlook Anywhere to work after the remote mailbox move from on premises to cloud?