An efficient combined numerical-analytical technique is developed for calculating states of the continuum spectrum in systems with quantum wells (QWs) with an arbitrary potential shape, described by a system of coupled Schrödinger equations, e.g., hole states in semiconductor QWs. Continuum-spectrum states are found exactly using the approach similar to the scattering theory. Scattering states (the in/out-solutions) and the S-matrix for the case of multichannel scattering in one-dimensional systems with QWs are constructed, and their symmetry is determined and analyzed. The method is applied to studying the hole scattering by GaInAs-InGaAsP QWs with strained layers. The hole transmission and reflection coefficients and the delay-time energy dependence are calculated in relation to parameters of the structures and values of the transversal momentum components. In the energy range in which the channel with heavy hole conversion into a propagating light hole is closed, scattering of the heavy hole on a QW has a resonant nature.