The St. Louis-based court on Monday rejected Zink's challenge of the drug process used in lethal injections.
The nation's high court is still weighing Zink's case, and Gov. Jay Nixon is reviewing Zink's clemency request.
Zink is scheduled to be put to death at 6 p.m. Tuesday for the killing of a 19-year-old Amanda Morton.
Fifty-five-year-old Zink was out on parole after serving 20 years in Texas on rape, abduction and escape charges when he abducted Amanda Morton after hitting her car from behind on a freeway ramp a mile from her Strafford home.
Zink later tied her to a cemetery tree in western Missouri, then snapped her neck before severing her spinal cord.