My feeling is things that are overwhelmingly likely do not get treated as information. For example, Harry's clothes go with him, but "Time" doesn't consider that to be information of his clothes still existing. It feels like that there's a Deus ex Machina aspect to how "Time" works and deals with information. Sometimes when you try to time-turn you just encounter Paradox.

So based on that I'd predict that if you try to time-turn with intention to get more uses out of the stone, you will encounter Paradox.