Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-wellpotential energy surfaces, often requires expensive quantumdynamics methods. Here, we explore the applicability andlimitations of the recently proposed single-Hessian thawedGaussian approximation for the simulation of spectra of systemswith double-well potentials, including 1,2,4,5-tetrafluorobenzene,ammonia, phosphine, and arsine. This semiclassical wavepacketapproach is shown to be more robust and to provide more accuratespectra than the conventional harmonic approximation. Specifi-cally, we identify two cases in which the Gaussian wavepacketmethod is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at thetop of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket hasenough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requiresonly a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. Wealso present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step