How do I continue a path with the Pen Tool?

There are multiple ways how to solve the problem. The fastest approach is using the Direct Selection Tool to select the two anchor points of the open path and join them using the Ctrl+J(PC)/Cmd+J(Mac) keyboard shortcut, or going to Object Menu>Path>Join.

You can add to an existing open path using the Pen Tool using a couple methods. If the open path is selected you should be able to switch to the Pen Tool and click on an end point of the open path. Or you can have the Pen Tool in use, hover over an unselected path and toggle Ctrl/Cmd to select the path and then append new points to the path. If Smart Guides are turned on an unselected path and its points will light up and allow the Pen Tool to add more points to it automatically.