Summary: In the last few years, several methods have been developed to deal with jump singularities in parametric or stochastic hyperbolic PDEs. They typically use some alignment of the jump-sets in physical space before performing well-established reduced order modeling techniques such as reduced basis methods, proper orthogonal decomposition, or simply interpolation. In the current literature, the transforms are typically of low resolution in space, mostly low order polynomials, Fourier modes, or constant shifts. In this paper, we discuss higher resolution transforms in one of the recent methods, the transformed snapshot interpolation. We introduce a new discretization of the transforms with an appropriate behavior near singularities and consider their numerical computation via an optimization procedure.