make switching accounts easy on office.com
By clicking account icon, you could make a drop down menu to switch accounts between business and personal etc.
Like on Google.com
I can sign in with my personal and work accounts at the same time, but if I do that when I go
to Office.com it always uses my work account, and the only way to switch to my personal account without signing out of the work account is go to Outlook.com and then click on a link there which takes me to Office.com. There should be an easy way to switch between personal and work accounts on Office.com; for example, you could have the option to switch account in the drop-down menu which appears when clicking on the account picture in the top right corner.