During an Exchange 2010 to 2013 migration, you may get into a situation where Outlook clients with mailboxes on Exchange 2010 can no longer connect to their mailboxes once the A record for Outlook Anywhere e.g. mail.domain.com is changed to resolve to Exchange 2013.
Although there are a number of causes, this may be because the RpcClientAccessServer parameter on the Exchange 2010 mailbox databases is set to the same FQDN as the Outlook Anywhere hostname. This means that once you change the mail.domain.com A record to resolve to Exchange 2013, the RPC/TCP clients will attempt to connect to the Exchange 2013 servers using RPC/TCP which won't work as this is no longer supported by Exchange 2013.
Usually Outlook Anywhere is enabled on Exchange 2010 but this is not always the case. If enabled, the clients *should* fail over to using Outlook Anywhere if they cannot connect using RPC/TCP on fast networks. If not enabled then clients will attempt to connect to Exchange 2013 using RPC/TCP and will fail. (Note that enabling Outlook Anywhere on Exchange 2010 is a required step in an Exchange 2010 to 2013 migration)
To resolve this issue, enable Outlook Anywhere on Exchange 2010 and force the Outlook clients to use HTTPS (Outlook Anywhere) on all connections. By default, on fast connections, Outlook will try to use RPC/TCP then HTTP and on slow connections Outlook will try to use HTTP first then RPC/TCP. This can be configured using Autodiscover by running the below commands from the Exchange Management Shell on Exchange 2010:
Set-OutlookProvider EXCH -OutlookProviderFlags:ServerExclusiveConnect
Set-OutlookProvider EXPR -OutlookProviderFlags:ServerExclusiveConnect
Once this is done, all Outlook clients on Exchange 2010 will be using Outlook Anywhere whether they are inside the office or not and you can change the mail.domain.com A record to resolve to Exchange 2013 so Exchange 2013 can proxy the request for the Exchange 2010 users.
Any clients that don't support Outlook Anywhere need to be configured to use Exchange 2010 by adding specifying mail.domain.com to resolve to the internal IP of your Exchange 2010 servers.
More information can be found here: http://blogs.technet.com/b/exchange/archive/2013/05/23/ambiguous-urls-and-their-effect-on-exchange-2010-to-exchange-2013-migrations.aspx