The association said there had been nine deaths in the jail, on the Isle of Sheppey, in 2014.
In November, Her Majesty's Inspectorate of Prisons (HMIP) said high-risk inmates were not being properly assessed and managed.
A Ministry of Justice (MoJ) said the inmate's body was found on Tuesday.
"An HMP Elmley prisoner was found unresponsive in his cell, at about 17:20 GMT," a spokeswoman said.
The HMIP investigation, which took place in June and was published in November, highlighted staff shortages and overcrowding at the jail.
The report said the prison holds 1,252 men, but was designed for only 985 inmates.
The MoJ spokeswoman said the latest inmate to die was pronounced dead shortly after paramedics arrived.
"As with all deaths in custody, the Independent Prisons and Probation Ombudsman will conduct an investigation," she continued.
"Every death is a tragedy for the individual and their families."
The MoJ confirmed nine people died at the prison in 2014.
Four were apparent self-inflicted deaths, four were deaths from natural causes and one has not been classified.
A man has been found dead in his cell at a prison in Kent - the third in as many weeks, the Prison Officers' Association has said.