Woman teacher is on a school outing, suddenly gets her period ten days before she should, sends kids off to look for mushrooms in the woods and takes herself deep into the forest to mop up with the handkerchiefs she has, because you always have handkerchiefs in Japan to dry your hands. Towels not provided in public washooms. Then she hides the bloody handkerchiefs where no one will find them and goes back to the group, still bleeding into her underwear, pesumably.
No she doesn't let go of those handkerchiefs. She keeps them stuffed in her underwear because seriously, Haruki, periods aren't a single effusion of blood that then stops. Don't you know any women at all?
You need bloody handkerchiefs for a plot point? Find something else then. Mattaku.