How could Haman be an Agagite if they were all wiped out by Saul?

Haman is described as an Agagite in the Book of Esther, which raises questions about his lineage since King Saul is said to have wiped out the Amalekites, including their king, Agag. However, it's possible that Haman's ancestry could trace back to remnants of the Amalekite people who survived Saul's campaign. Additionally, "Agagite" might refer to a symbolic or cultural identity rather than a direct descent from King Agag, allowing for the inclusion of Haman in this identity despite the earlier extermination.