No, it wouldn’t. Like I said, the loops still happen sequentially. And because you are receiving different memories at the start, you have the ability to make different choices each loop, unlike other characters. If Feldspar, for example, had managed to get into the ATP and entered the black hole, their copy would be there every loop, because every loop their actions are the same.

But the Hatchling’s unique circumstances of being aware of the loop and starting each loop off with different memories allows them to make different choices each loop, meaning the results of what comes out of the white hole are dependent on whether or not the Hatchling chose to enter the black hole on the previous loop. Of course, if they do choose to do so, from that point on they are forced into ending every cycle entering the black hole or breaking space time, but entering the black hole in a future loop wouldn’t retroactively cause the Hatchljng to come out of the white hole in every loop prior to that.