Spine & Neuro Center. Governors Medical Tower, 201 Governors Drive, first floor • Huntsville, AL 35801. (256) 533-1600 • Toll free: 1 (888) 736-1762. Hours of service: Monday - Friday, 7 a.m. - 3:30 p.m. Huntsville Hospital Spine and Neuro utilizes a unified and comprehensive approach when it comes to caring for our patients..