The Hat-and-scarf girl can be found in the Block World. She teleports Madotsuki to the top of a random block that is normally inaccessible. Some of these have a portal to the White Desert. On the return trip, Madotsuki is teleported to another random spot in the Block World unless she interacts with the Hat-and-scarf girl as a snowman, in which case she teleports back to the girl's side.
The Hat-and-scarf girl appears to be invisible except for what she is wearing: a hat and a scarf. If you use Stoplight she and her outfit appear completely, revealing her as a girl.