What was the Velvet Scare during J. Edgar Hoover's time, and how does it relate to The Crucible?
I see that you are asking about J. Edgar Hoover's era on this question, and so I conclude that you are thinking the Velvet Scare is something that happened during his lifetime.
I have taught US history for years at the high school and college level and I have never heard the term "velvet scare" in connection with J. Edgar Hoover (or in any other context, actually). I really wonder if you might be talking about the Red Scare.
Hoover was connected to the Red Scare and it is connected to The Crucible. The Red Scare was a time in the '50s when Americans were very worried about communists in the US. Hoover and the FBI kept files on people suspected of being communists and such people were often persecuted. People often say that this was a witch hunt similar to the one in Miller's play.