Sam, the Beis Yosef in Hilchos Tzitzis brings the Gemara about the Sudar as a Mekor for the Middas Chassidus of a covering besides for the anti-Gilui-Rosh device.
WIY, his point is that its only merit is that it is the Derech to wear it. That makes it important. Those who never wear a hat don’t relate to it as something for Davenning.