An elegant hook bijection (perhaps, THE hook bijection) was found by Pak and Stoyanovskii [1]. It avoids the involution principle. It works on the basis of jeu de taquin moves which are recorded in a beautiful manner. However, [1] does not contain a proof that the algorithm works. A proof is provided in [2]. Using similar ideas, Novelli and Pak [3] claim to have found a bijection for the shifted hook formula that avoids the involution principle.
[1] I. M. Pak and A. V. Stoyanovskii, A bijective proof of the hook-length formula and its analogues, Funct. Anal. Appl. 26 (1992), 216-218; translated from Funkt. Anal. Priloz. 26 (No. 3) (1992), 80-82.
[2] J.-C. Novelli, I. M. Pak and A. V. Stoyanovsii, A direct bijective proof of the hook-length formula, Discrete Math. Theoret. Computer Science 1 (1997), 53-67.
[3] J.-C. Novelli and I. M. Pak, in preparation.