1. What is respectable garb differs according to time and place. Should Sephardim today wear fezes because at one time it was the mark of a man of standing?
2. Many Orthodox men did not wear kippot at work or while engaging in secular studies. In fact, in Rav Hirsch’s school, the boys studied secular subjects bareheaded. In fact, originally it was only a middat chassidut. If you see old passport pictures, even talmidei chachamim have their heads uncovered as it was a requirement of the government (US passport regulations still say this but there is a religious exemption). The Cahfetzchaim also wore a cap as did Rav Kook when he was a rav in Latvia (after he came to Yerushalayim he changed to a spodik). I heard that Rav Moshe wore a straw hat in his bungalow colony