and it also says not with the KOVEA KOTON SHEHTACHAS HOKOVEA. it seems that the Mishna Berurah understands that the there a two coverings, a small one under the big one and both were worn at the same time.
Eh. Based on the context, I would say that’s a weak proof. He just means not the small kova that is commonly worn under a hat. If there was a specific inyan of two coverings, he would have said it straight out.
I’ve seen rabbonim take off their yarmulkes as they put on their hats for davening.