Ike never wore a hat.. and I cannot imagine Harry Truman in a top hat so I did not bother to look for an image of his inauguration. But what the hell..
I looked and damn, there he and VP Barkley both with top hats.
"Eisenhower was the first president to eschew the top hat and tails. . . . "
Truman, IIRC, ran a haberdashery before he got into politics. As such, he was very familiar with hats from both the seller and the customer’s viewpoint.