It depends on the specific story really.
However, if you're twisting my arm , I'd either go to the 30s or the 60s.
The 30s would be much more stark since this would be the pre- MLK, Snick, Nation of Islam etc, civil rights period .
There would truly be a fierce response to a Black Superman to the world of 1938 . That was the time of the depression, the resurgence of the Klan, segregation , the rise of Fascism worldwide, Jessie Owens at the Berlin Olympics, etc.
That world dealing dealing with a Black Superhero who's the most powerful being on the planet would present a compelling, yet very dark portrait.
In the 60s, you have several different civil rights and social changes going on .
So while there's MLK, Medgar Evers, Snick, The NAACP, Brown v Board, Boycotts etc, there's also the growth of women's rights, anti war protests , social and cultural revolutions, and the divide between The Greatest Generation and their Baby Boomer children.
There's alot more to chew on and contend with in terms of other factors which complicate the picture in the 1960s than the 1930s imo.
That's why I think it's easier to tell a more focused and contained story in the 30s than in the 1960s. The 30s certainly had its complications for sure, but the 1960s was like 20 years worth of changes and events that occurred in one decade.
Either way, if it were up to me, I'd probably stick to the early to Mid 20th century.