While in the Personal Details screen of the record, you have a few tabs at the top. The second of those tabs is the Address Details tab.

Clicking on the Address tab will bring you to the Address Details overview for that record. You will see an individual pane for each address that is listed for that record.

At the top right corner, you can click on the + button to add a new address. Fill in the details for the new address, then click on the save and close button to be brought back to the Address Details pane.

Within each of the address panes, you have two buttons. First there is a clock icon button that you can click on to see the history of the address. Also there a three dot button where you can choose to edit or delete the address.