There was a time and place for men’s accessories like dress hats, walking sticks and pocket watches. In modern America, there’s little place for these — modern style has, for the most part, moved on.

It was a very special moment in any young man’s life: buying his first pipe. On a recent Saturday afternoon, 25-year-old Ben Darge stood at the counter of Iwan Ries, which bills itself as America’s ...