The court unanimously granted the stay to determine whether a recent change to Georgia's lethal-injection protocol violates state law, The Atlanta Journal-Constitution reported.
Hill, 52, is on death row for bludgeoning to death a fellow inmate in 1991, when he was serving a life sentence for the 1985 death of 18-year-old girlfriend.
Advocacy groups for the developmentally disabled have been trying to stop Hill's execution. Hill has an IQ of 70 and two judges have ruled Hill is probably not mentally disabled.
Hill had already eaten what he thought was his final meal Monday night when he learned his 7 p.m. EDT execution had been called off.
"I'm just profoundly grateful the Supreme Court granted this stay," Brian Kammer, one of Hill's lawyers, was quoted as saying. "A terrible miscarriage of justice was avoided, for now."
The newspaper said it could be months before the state Supreme Court decides Hill's appeal.
Hill was initially scheduled to die last week by lethal injection but his death was delayed after the Department of Corrections said last Tuesday it was switching to a single drug process from the three-drug cocktail previously used.
Notable deaths of 2014 [PHOTOS]