The sound most like the egg song is a contact call. Such pullets for me are no longer staying in the tight groups with broodmates. It may be the time they would normally leave to join other groups where they would settle down and breed.
They makes several calls, which are quite similar and don't have anything to do with egg laying, per se. If they get separated from the group, they'll stand and call. If they hear or see something, they give warning, and stand and call. If another pullet seems to do something they don't care for or seems to act out of the order, they'll stand and complain loudly. Geeeesh. It really doesn't take all that much for a some of them to sound off. Others, are virtually silent, confident and relaxed all the time. It just varies.