Previously reported measurements of the NMR spin-lattice relaxation time of $^{59}\mathrm{Co}$ in ${\mathrm{K}}_{3}$Co${(\mathrm{CN})}_{6}$ have been extended towards lower temperatures ($T<100$ K) to determine the contribution of the low-frequency normal modes of vibration of the Co${(\mathrm{CN})}_{6}$ octahedra to the relaxation. It has been found that the temperature dependence of the relaxation in the region $35<~T<~300$ K can be reasonably well accounted for in terms of the individual contributions of three intermediate-frequency modes (380, 414, and 565 ${\mathrm{cm}}^{\ensuremath{-}1}$) and two low-frequency modes (104 and 129 ${\mathrm{cm}}^{\ensuremath{-}1}$) and that overlapping of closely spaced modes tends to worsen the agreement with experiment.