Outlook 2016 and outlook.com

Revamping Outlook.com emails (e.g. xxx@msn.com,  xxx@hotmail.com like emails) requires reconnection.  

Follow the instructions to reconnect.  You could delete the old outlook profile and create a new one.   Deleting the profile does not delete outlook pst files corresponding to existing email accounts.  You will be prompted to add your Outlook.com account.  The process is straightforward except if you have enabled two-step verification for your Microsoft account that requires using an App password instead of your Microsoft account password to set up the email.   You will see only your Outlook.com email account when you open Outlook after creating the profile.  You can easily add all other emails back (File > Open & Export > Open Outlook Data File >Select an email account's pst file).

Following instructions to add an Outlook.com to Outllook 2013/2016 desktop app may appear successful, but the synchronization may not work, especially the contacts.

The quickest way to migrate old emails data to a new instance is starting from scratch, then using existing file.

1. Create a new profile with one email address, and always use that profile.
2. Import that data from the old pst file this email to this account. 
3. Add other emails one by one. IMPORTANT: on the POP Account Setttings page, choose "Use an existing data file".  Move the old pst file a desirable location. 

It takes less than 10 minutes to set OL 2016 for half a dozen emails starting from scratch and inheriting old data.