Women's role in The Odyssey is interesting. On the one hand, the female characters in this quintessential epic poem display more depth than other female characters in epic poetry. Calypso, for instance, is a powerful woman able to hold sway over an epic hero like Odysseus. Along the same lines, Athena holds a prominent place on Mt. Olympus and appears to be able to get her way in most scenarios by swaying the opinions of Zeus. In this respect, then, it seems like women in The Odyssey have considerable power, and this fact could have reflected growing independence for women in Ancient Greece.
However, while women in The Odyssey might be more powerful than their literary predecessors, they are still largely subordinated to their male counterparts. Penelope's role, for instance, is to wait at home while Odysseus goes wandering, and Athena, as powerful as she is, seems to exists largely to help Odysseus escape trouble. As such, women in Homer's poetic world might not be as independent as they seem, and this fact certainly reflects the state of women in Ancient Greece, as Greek women were largely forced to depend on men and had little autonomy outside of the home. In that case, if women's independence was growing during the composition of The Odyssey, they still most likely did not have much power in Ancient Greek society.