An increasing number of applications, including non- or minimally invasive diagnostics and treatment as well as various cosmetic procedures, has resulted in a need to determine the optical properties of hair and its structures. We report on the measurement of the total attenuation coefficient of the cortex and the medulla of blond, gray, and Asian black human scalp hair at a 633-nm wavelength. Our results show that for blond and gray hair the total attenuation coefficient of the medulla is more than 200 times higher compared to that of the cortex. This difference is only 1.5 times for Asian black hair. Furthermore, we present the total attenuation coefficient of the cortex of blond, gray, light brown, and Asian black hair measured at wavelengths of 409, 532, 633, 800, and 1064 nm. The total attenuation coefficient consistently decreases with an increase in wavelength, as well as with a decrease in hair pigmentation. Additionally, we demonstrate the dependence of the total attenuation coefficient of the cortex and the medulla of Asian black hair on the polarization of incident light. A similar dependence is observed for the cortex of blond and gray hair but not for the medulla of these hair types.