The FBI pretty much was incorruptible until they were forced to lower their standards. After that, they went downhill.
Are you referring to Agent hiring after Hoover's death? Because Connolly was hired when he was still alive. this is from Wikipedia:
"On August 1, 1968, U.S. House Speaker John William McCormack wrote a personal note to J. Edgar Hoover on behalf of a constituent. The letter began, "Dear Edgar, It has come to my attention that the son of a lifelong personal friend has applied to become a special agent of the Federal Bureau of Investigation..." He was appointed to the FBI in October 1968. FBI Supervisor John M. Morris, who would also face charges of corruption, was Connolly's supervisor during much of his time working for the FBI."
