According to the Shaafi'i school of thought it is a Sunnah to drink Zam Zam water at any time and not only specifically at the end of the Tawaaf. And it is a Sunnah, in their opinion, for everyone to drink Zam Zam water even if the person is not performing Hajj or Umrah.
Kuwaiti Encyclopedia of Fiqh
Volume 24 Page 14
You may also find the following video regarding Zam Zam water interesting.