Among the raft of reissues from heritage imprints, this one stands out, partly because the author’s writing career lasted little more than six months before her death in a motorcycle accident in 1966 at the age of 22. Diane Oliver published four stories in her lifetime and two more posthumously; they are all included here along with eight stories previously unpublished.

Oliver’s subject is the black female experience in 1960s America, in the period when racial segregation was illegal but prejudices remained ingrained – but the tales succeed for their literary qualities, not their subject matter. The title story follows a family preparing for their son to be bussed to a newly integrated “white” school and their fears both from those attacking them (should they burn abusive letters, or keep them as evidence?) and those trying to help: cruising police cars intended to reassure them just make them more nervous.

The characters are not warriors for justice but ordinary people, who sometimes want a quiet life. “Now sooner or later they’re going to integrate everything. I know that,” says the father of the family in When the Apples Are Ripe. “But sensible people with a future don’t get involved.” And Oliver’s techniques keep the stories varied even while the themes remain linked. There’s dry humour in The Closet on the Top Floor, where a black woman attends a college with white students, but rejects her mother’s suggestion of joining the drama sorority because she “didn’t see how she could play the maid’s part for four years”. In Banago Kalt, a woman visiting Switzerland is beset by fatuous demands. “Was she in any danger of being lynched, they wanted to know.”

The previously unpublished stories are of varying quality. No Brown Sugar in Anybody’s Milk is one of the best overall, with an unexpected revelation that deepens its emotional resonance. Others, if published in Oliver’s lifetime, would have benefited from an editor’s hand, such as the overlong impressionistic experiment Frozen Voices, or Our Trip to the Nature Museum, an unsubtle story of a teacher involved with a black child’s home life. We can only imagine what wonders Oliver might have produced had she lived, but the precocious talent on display here is cause enough for celebration.

• Neighbors and Other Stories by Diane Oliver is published by Faber (£9.99). To support the Guardian and Observer order your copy at guardianbookshop.com. Delivery charges may apply