In Megillas Esther 3:1 Haman is called the Agagite which I always assumed meant he was a descendant of Agag the king of the Amalekite in 1 Shmuel 15:8. His lineage is confirmed in Megillah 13a as being a genetic descendent of Agag the Amalekite.
ומה שילם לי ימיני דלא קטליה שאול לאגג דאתיליד מיניה המן
If the point is to tell us that Haman was an Amalekite, why not just say so explicitly?