I was rarely in that situation although I did think it was stipid for women to grab the tea towel. That was rather insulting (I thought) If you were going to be serious about it then the least you could do is carry a small scarf with you but I don't think any JW woman did.
I thought it was totally stipd though when an elder told me that if I was going to teach sign language to a brother I needed a head covering. I had been teaching them for years and no one said anything. Then all of a sudden it was an issue. Stupid Just plain stupid.