Using valid experimental parameters, we quantify the magnitude of resonantly phonondriven precession of exchange magnons in freestanding ferromagnetic nickel thin films on their thickness L. Analytical solutions of acoustically driven equations for magnon oscillators display a nonmonotonous dependence of the peak magnetization precession on the film thickness. It is explained by different L-dependence of multiple prefactors entering in the expression for the total magnetization dynamics. Depending on the ratio of acoustic and magnetic (Gilbert) damping constants, the magnetization precession is shown to be amplified by a Q-factor of either the phonon or the magnon resonance. The increase in the phonon mode amplitude for thinner membranes is also found to be significant. Focusing on the magnetization dynamics excited by the two first acoustic eigenmodes with p = 1 and p = 2, we predict the optimum thicknesses of nickel membranes to achieve large amplitude magnetization precession at multi 100 GHz frequencies at reasonably low values of an external magnetic field. By extending the study to the case of Ni-Si bilayers, we show that these resonances are achievable at even higher frequencies, approaching the THz range.