This thread by u/efini lists a number of people who claimed to be the last former living slaves. Although many of the claimants have unverifiable claims, you'll see that Peter Mills's census records do support his age claims - he was born in 1860, three years before emancipation and died in the 1970s, at least five years after the passage of the Voting Rights Act.

So the answer appears to be that, yes, there was at least one person for whom there is good evidence who was born a slave and died after the major Civil Rights legislation was passed in the 1960s.