In Tzitzit Halacha LeMa'aseh (Chapter 9, footnote 12), The Leket HaKemach HaChadash (8:16) Is quoted as saying:
The custom in Germany is that the whole congregation only covers their head (with the Tallit) while saying Shema on Yom Kippur and Yom Kippur Katan. However, many of the G-d fearing put a hat on top of their Kippah if they are not covering their heads (with the Tallit).
I wasn't able to find a copy of the sefer to read it inside, so I'm not entirely sure the context, but that would seem to be a source for putting on a hat when the Tallit is taken off the head.
This appears to be the Sefer, by R' Yaakov Tzi Katz, but the relevant section is not part of the free preview.
The question then becomes, why remove the Tallit from one's head at the end of Mussaf.
There is a Sicha from the Previous Lubavitcher Rebbe from the second night of Chol HaMo'ed Sukkot 5697 (1936), where he mentions that there is a Kabbalistic reason to remove the Tallit from one's head by the Ein Ke'lokeinu prayer (which immediately follows the Chazan's repetition of the Mussaf Amidah). [Chapter 12 footnote 65 of Tzitzit Halacha LeMa'aseh points out that this does not appear to be the Chabad custom]
So to put these two ideas together. If one deliberately removes the Tallit from his head by Ein Ke'lokeinu, but wants to have another covering on his head when the Tallit is removed from his head, he will have to go get his his hat before the chazzan concludes the repetition of the Mussaf Amidah.
I have no idea if this is what the members of the synagogue in your question have in mind.