There are several underlying context clues within "Everyday Use" that reveal the time period to be sometime in the late 60s to early 70s. Each of these clues comes the night Dee comes home with her new boyfriend "Hakim-a-barber."
In the late 1960s and early 1970s, in the wake of the peak of the Civil Rights Movement (and leaders such as Martin Luther King and Malcom X), many African Americans began exploring and embracing their African Heritage. It became known as the "Black Power" movement, or Black Nationalism, and was done as a means to bolster self-esteem and create a sense of pride and unity that was truly unique.
Aside from the description of Hakim-a-barber's "wild looking hair" (indicating he has an afro), another indication that Dee and Hakim-a-barber are involved in Black Nationalism is the changing of their names. Dee wishes to become Wangero Leewanika Kemanjo. This shows that she and her boyfriend are part of the Black Muslims, a group (originally led by Malcom X) which believed that white society was incapable of being non racist. Therefore, it was in African American's best interest not to integrate, but rather, to remain separate and grow in power independently. The act of changing their names was a symbolic rejection of their given "slave names" or "Christian names."
There are also smaller clues alluding to the Black Arts Movement and the Black Feminist movement in the short story. Walker's use of women as the the focal points to this story is one clue. These women fighting over ancestral pieces of value (the quilts) is another.
dee doesnt want her quilts anymore