A majority of the justices in November cleared the way for the execution of Smith, sentenced to death for his role in a 1988 murder-for-hire plot.
The method in Alabama posed an "intolerable risk of torture, cruelty or substantial pain," Smith's lawsuit stated.
A judge dismissed Smith's lawsuit, but on Nov. 17, the day of his scheduled execution, the Atlanta-based 11th U.S.
The 11th Circuit also in a separate decision stayed his execution, but after the state appealed to the Supreme Court, the justices allowed it to proceed.
Smith's case is not a challenge to the death penalty itself.