The word HAT is contained in the wording of the clue,but quite how this is applicable to Holy Willie I don't know.
By the way he wasn't a priest merely an elder.
The only thing I found helpful was to Google images for the said character, and there appears to be a theme. Still not sure of the correct answer though.
for Leslie - I'm pretty certain it wouldn't have been a cassock (Burns would have died laughing) more likely a sash, but I suppose you need to know the west of Scotland to understand that.
Please forgive a sassenach's ignorance; even more inexcusable since I slept in a room in Kirkcudbright previously occupied by RB - or so the plaque on the wall would have it.
You are perhaps coming towards the answer here John.
I don't think it is a sash but I would go for APRON,as worn by free masons,which it would appear that Holy Willie was one.