The current method is first person to touch the site at midnight will trigger the event ...
But if they are already on the site the site does not re-update with each click ...
What you may be looking for is a method to know Joe hits the site at 11:59pm ...
This hit runs the update on specials ...
Joe continues to shop for 30 minutes ... the update is not run again it was ran by Joe at 11:59pm once per session ...
However, Harry hits the site at 12:01am ...
This hit runs the update on specials ...
Now both Joe and Harry see the new Specials ...
If you need a secondary timer where if the session already exists and switches across midnight ... this could be customized so Joe does not see "yesterday" prices until Harry comes in ...