It is unclear what you mean by "reopen" the path.
If the path is closed and you want it to be an open path, you can either cut it with the Scissor tool, or direct select any bezier segment and delete it.
However, the mention of adding points suggests that you may not mean making a closed path become open.
If the path is already an open path and you want to continue appending to the end of it, you can click the Pen tool on either endpoint to make new anchor points, or as Philip suggests, you can start dragging the Pencil tool near one of those endpoints.
If neither of the above is what you mean, can you describe in more detail what you are trying to do?