Is it Sunnah to wear a Muslim hat?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Ibn al-Qayyim while describing the Prophet's way of dressing, mentioned in his book Zaad al-Ma'aad “that the Prophet used to wear a turban called as-Sahaab; he presented it to ‘Ali .” He added: “The Prophet was used to wearing such a turban and there was a cap under it. Sometimes, he used to wear the cap without a turban, and vice versa.”

So, we deduce that it is from the Prophetic practice to cover his head and such a practice is not necessarily to be with a cap or a turban or both.

Besides, 'Amr ibn Hurayth narrated: “I saw Allah's Messenger on al-Mimbar (pulpit) wearing a black turban, letting down its ends between his shoulders.” [Muslim]

To conclude, it is a Prophetic Sunnah for men to cover their heads with caps and the like.

Allah knows best.