The justices on Thursday will take up the case of 57-year-old Michael Dale St. Clair, who was convicted in the 1991 slaying of distillery worker Frank Brady in Bullitt County.
St. Clair has won three trials in the case, which has lingered for years in appeals.
St. Clair and another inmate escaped from an Oklahoma prison before going on a multistate spree that ended in Kentucky with Brady's death. St. Clair also faces a murder charge in New Mexico for the 1991 kidnapping and slaying of paramedic Timothy Keeling.
St. Clair also received a second death sentence for capital kidnapping from the Hardin County Circuit Court.