Airborne radar sensors capture the profile of snow layers present on top of an ice sheet. Accurate tracking of these layers is essential to calculate their thicknesses, which are required to investigate the contribution of polar ice cap melt to sea-level rise. However, automatically processing the radar echograms to detect the underlying snow layers is a challenging problem. In our work, we develop wavelet-based multi-scale deep learning architectures for these radar echograms to improve snow layer detection. These architectures estimate the layer depths with a mean absolute error of 3.31 pixels and 94.3% average precision, achieving higher generalizability as compared to state-of-the-art snow layer detection networks. These depth estimates also agree well with physically drilled stake measurements. Such robust architectures can be used on echograms from future missions to efficiently trace snow layers, estimate their individual thicknesses, and thus support sea-level rise projection models.