“I know that my family has had people (more than one) not trust our Kashrus before because my father doesn’t wear a hat.”
I wasnt accepted into a mesivta because my father was wearing a straw hat (this was in 1983). (this was confirmed when we asked an uncle of mine who had three sons in the yeshiva to ask….he asked why his neighbors son wasnt accepted into the yeshiva and they told him that they were able to tell by the fathers hat that we werent their kind of people – they officially told us that they didnt have enough beds in the dorm!).
As for the hat, some regiments in the army of hashem include a black felt fedora as part of their uniform, while others do not. Within these regiments, the styles vary by unit. Some units have pinches in their hat, some dont, others choose the wide brimmed style, while others the small, some go brim down and some brim up, some perched way back on their head and others pulled down over their eyes in the front.