© 2017 American Physical Society. We extend our recent result for the spin-relaxation time due to acoustic electron-phonon scattering in degenerate bands with spin mixing [New J. Phys. 18, 023012 (2016)NJOPFM1367-263010.1088/1367-2630/18/2/023012] to include interactions with optical phonons, and present a numerical evaluation of the spin-relaxation time for intraband hole-phonon scattering in the heavy-hole (HH) bands of bulk GaAs. Comparing our computed spin-relaxation times to the conventional Elliott-Yafet result quantitatively demonstrates that the latter underestimates the spin-relaxation time because it does not correctly describe how electron-phonon interactions change the (vector) spin expectation value of the single-particle states. We show that the conventional Elliott-Yafet spin relaxation time is a special case of our result for weak spin mixing.