The jewel appears weaker than later in the series; suggesting it was still recovering its strength after being freed from Kagome. Considering Inuyasha could kill Mistress Centipede easily.
Unlike the manga, Kagome has a distinctly different face than Kikyo. So it's more laughable Inuyasha mistakes her for Kikyo.
It makes Kagome didn't emerge from the well at the same time as Mistress Centipede. The Demon fell before her into the time travel void; meaning it left the well earlier.
Judging from where the Jewel came out of Kagome's body, it was somewhere between her intestines and ovary.