We're planning to upgrade our Exchange 2010 server to 2016. As Exchange database is big in size so both the version will co-exist for a month or two during phased mailbox migration. Currently, HC is installed on Exchange 2010. 
Please send me the document on how to migrate HC from one Exchange server to another and how will HC work in case of co-existence.

You can add both Exchange  2010 and 2016 in HC10 panel. You can complete the migration from Exchange 2010 to 2016. After this contact our support team by sending an email to support@hostingcontroller.com and they will further guide/assist you in pointing the Exchange Organizations , Mail Domains, etc to the newer Exchange 2016 version.