Another feature we have at our disposal in Office 365 is the online archive feature. Archiving is a place where you could store an unlimited amount of email and keep them separate from your normal email folders. Emails stored in the archive are accessible as a separate set of email folders from the desktop version of outlook and OWA, but do not count toward your 100GB email storage limit.
Additionally, you can set up archiving policies to automatically archive emails older than 1 year, 2 years, 3 years, 4 years, or 5 years automatically.
The University of Kentucky has a nice information sheet explaining some details about online archiving and how to set up archiving policies in OWA and Outlook 2013/2016.
The online archive is not turned on by default. It may have been turned on for some during the initial migration if you were already archiving locally. If you do not have it turned on and want it turned on, contact Karl Lindauer or your IT support personnel.