Exiting Schengen and reentering does not seem to be a requirement to make your stay valid.
[EDIT] However, I just found that the French Consulate in Sydney specifically says you should leave Schengen and reenter, suggesting to go to the UK :
If you want to stay in the Schengen space (for up to 90 days) at the expiry of your working holiday visa, you will have to leave France and the Schengen space and re-enter the Schengen area the following day as a tourist for 90 days within a 6 months period. You may leave the Schengen area (passport stamped at the border) by going to the UK for example.
As other answers show it, it is possible to spend up to 90 days inside the Schengen area for tourism purpose even if you had a long-stay visa before.
This is only experience and is not an authoritative source, but a friend (who does not need a visa for short stay) recently ended her stay on a student visa in France. She decided to stay for tourism afterwards. She did not leave the Schengen until more than 10 days after her student visa expired (with an exit stamp showing the date). She entered Schengen area and exited twice soon after, and never had any issue with immigration (who barely checked her visa the second time). She crossed the border at different places in Europe, making it relatively obvious she was traveling (and not working for example).
Therefore, even though the Consulate of France in Sydney says otherwise, as quoted above, it appears likely that exiting and reentering to have exit and entry stamps showing the stays are separate is not a requirement. It seems possible to have continuously a long-stay visa and a short stay (if no short-stay visa is necessary), but I would recommend to leave Schengen if possible. I suppose that any proof that you are not working (e.g. a document showing the end of a contract) would help, in case the border police is suspicious.