The Chanukah story actually began with the historic meeting between Alexander the Great and the Kohein Hagadol, Shimon Hatzaddik. Alexander dreamed of this meeting and was very impressed with the holy garments worn by the high priest.
He promised that he would act kindly with the Jews and not do them physical harm. He was respected by the Jews to the extent that Jews are still called Alexander, to this day.
Nevertheless, Alexander did cause great harm to the Jews, with his introduction of Greek culture into Israel. Many Jews became Hellenists and abandoned their observance of Judaism in order to emulate the Greeks.
This has been the story of the Exile. Either we needed to devise tactics to overcome a physical threat to us, as in Shmad. Or, we needed to overcome a spiritual threat when we were enticed with foreign cultures.
Our experience with the Greeks provided a challenge on both fronts. We fought a physical war with the Greeks and we tried to restore our spirituality with the re-dedication of the Temple. This is the dual celebration of the Chanukah holiday.