We present a numerical study of the interaction quench dynamics in a superfluid ultracold Fermi gas confined in a three-dimensional cigar-shaped harmonic trap. In the present paper, we investigate the amplitude mode of the superfluid order parameter after interaction quenches, which start deep in the BCS phase and end in the BCS-BEC-crossover regime. To this end, we exploit the Bogoliubov–de Gennes formalism, which takes the confinement potential explicitly into account and provides a microscopic fully coherent description of the system. We find an anharmonic nonlinear oscillation of the modulus of the superfluid order parameter, i.e., of the Higgs mode. This oscillation persists for large times with only a small amplitude modulation being visible. We connect the frequency and the mean value of this oscillation with the breaking of Cooper pairs in the superfluid phase. Additionally, we demonstrate that the occurrence of this persistent oscillation is connected to the onset of chaotic dynamics in our model. Finally, we calculate an interaction quench phase diagram of the Higgs mode for quenches on the BCS side of the BCS-BEC crossover and discuss its properties as a function of the aspect ratio of the cigar-shaped trap.