According to the Talmud (Megillah 16a), the verse describes Haman as having a covered head because he was covered with embarrassment.
The Me’am Loez adds that Haman did not want people to recognize him, so he had his head covered.
The Ben Ish Chai writes that there was a custom then in Persia to wear large-brimmed hats. Haman’s daughter’s waste knocked Haman’s hat off his head, ruining it. The gallows he unintentionally built for himself were calculated exactly to fit him with his hat, in order for him to be recognizable. In order to account for this otherwise extra half ama, his hat was nailed to his head on the gallows in order to show everyone that even big advisers are subject to the king’s decree.
According to Ben Ish Chai, Haman’s uncovered head was a reward for Shimi’s wife saving the righteous Achma’atz and Yonasan from Avshalom (Shmuel 2 17:19) by uncovering her hair. Mordechai, being Shimi’s (and his wife’s) descendant, was rescued from Haman’s evil designs in the merit of his ancestor’s pious act.