Some years back, there was a fire that damaged an apartment building that housed Somali families. The then mayor of Minneapolis did cover her head when she visited the sad area.
I don't think anyone gave her a hard time.
In a similar manner, women who visit with the Pope do cover their heads, and wear veils, I think, too, even if they are not Catholic. I remember former Israeli prime minister Golda Meir met with the Pope and said she looked to the Kennedy women to see what they wore.
Orthodox Jews do seek modesty and will not meet with a woman with exposed arms and a short skirt and exposed cleavage.
Why not show some respect?
about Golda
At first she planned to wear a black shawl because she had seen the Kennedy sisters wearing them at the United Nations when the Pope visited there. The Israeli Ambassador to Rome, Amiel E. Najar, told her that only Catholics wore such shawls, so it was decided that a black hat would be suitable.
But Mrs. Meir, who had no black hat, did not want to buy one because she could not use it afterward. Then, Maariv said, she remembered that she had one at home, left over from old state events, and had it flown to Rome.
https://www.nytimes.com/1973/01/20/archives/mrs-meir-says-tension-marked-talk-with-pope-mrs-meir-says-that.html