Inuyasha understood that Byakuya could have killed Kagome when he attacked. He tried to divine Naraku's true purpose. Why use a delayed meido on Kagome instead of killing her outright? Why go to the well? Why make the well disappear?
Inuyasha also understood that the jewel was after Kagome and kept her alive for some reason, even though it feared her. He begged Tessaiga to find the same meido that Kagome was in and trusted the sword to do it.
Inuyasha's meido connected to Kagome's end of the well, meaning the jewel had taken her to the future through the well and then closed the exit.
Inuyasha has started to show some very complex thinking. He was the first to realize that he wasn't the only one protecting Kagome, at least from Naraku. He also seems to have begun undrstanding that it is impossible to fully protect anyone.
We also see that he had completely united with his sword and seems to carry on psychic conversations with it. He knows that the sword possesses either knowledge or sight that he does not; and he trusts it to find what he cannot, in this case Kagome's location.
It's ironic that the jewel took Kagome to her own time. Does this imply that the jewel now belongs there as well? It seems certain that it intends to begin its new existence in the 21st century. Perhaps this is proof that the jewel has its limits. For example, it can't change history and has to continue its existence from the future where it re-entered the world.
This also implies that the jewel was lying. It does not have the power to turn either Inuyasha or Naraku into full and perfect demons. There is also something of an implication that the circumstances leading up to Inuyasha's demon aspecte emerging were possibly orchestrated by the jewel, i.e., have Naraku keep attacking Inuyasha with detachments until Tessaiga broke and Inuyasha became desperate enough for his demon side to take over. Was that how the jewel grants a wish? The only time Inuyasha had the complete jewel in his possession he intended to use it to become full demon. That was his wish. And maybe the jewel granted it, in a roundabout way.
On thing is certain, after Tessaiga broke, Naraku never directly attacked with another detachment until he sent Kanna after the group; and the goal then seemed to be Inuyasha's demon side. It was either kill Inuyasha or leave him as a mindless murderous beast.
This also fits in with the history of anyone trying to use the jewel becoming unhappy, or paying for it with their lives. After all, Inuyasha was "killed" right after announcing to the jewel what he wanted; and even Naraku refused to use the jewel's full power until he was desperate.