25-year-old Dashawn Carter was taken into custody on May 5 on burglary charges. He died just two days later.
The cause of Carter's death is under investigation.
A federal monitor has given the city until the middle of May to correct issues at the jail or be placed under federal control.
Sixteen people died while being held in custody at Rikers in 2021.
Rikers Island has been a major issue for lawmakers in New York City for the last several years. Plans to close the facility have moved slowly, while the average length of stay in the city's jails has increased from 187 days to 329 days.
Last month, new Department of Correction Commissioner Louis Molina was grilled for hours by City Council members angry over the agency's alleged mismanagement of the jail.
As crime rises in New York City and police make more arrests, issues at Rikers may become more exacerbated.
Last year, the city announced that it was bringing in private security to help deal with staffing shortages at the jail.
However, advocates continue to call for the jail to be shut down entirely, with Public Advocate Jumaane Williams calling the conditions there a "humanitarian crisis."