The remagnetization process after ultrafast demagnetization can be described by relaxation mechanisms between the spin, electron, and lattice reservoirs. Thereby, collective spin excitations in form of spin waves and their angular momentum transfer play an important role on the longer timescales. In this work, we address the question whether the magnitude of demagnetization - the so-called quenching - affects the coherency and the phase of the excited spin waves. We present a study of coherent magnetization dynamics in thin nickel films after ultrafast demagnetization using the all-optical, time-resolved magneto-optical Kerr-effect (tr-MOKE) technique. The largest coherent precession amplitude was observed for strongly quenched systems, indicating a well-defined precession phase for all pump pulses at a demagnetization of up to 90% in this system. Moreover, the phase of the excited spin-waves in Ni increases with the pump fluence, indicating a delayed start of the precession during the remagnetization. We compare these findings to recent studies in Ni80Fe20 (permalloy), to evaluate the influence of the magneto-elastic coupling and non-linear spin wave dynamics on the magnetization dynamics.