  We report a comparative analysis and theoretical description of the
superconducting properties of two spin-valve-valve structures containing the
Heusler alloy Co$_2$Cr$_{1-x}$Fe$_x$Al$_{y}$ as one of two ferromagnetic (F1 or
F2) layers of the F1/F2/S structure, where S stands for the superconducting Pb
layer. In our experiments we used the Heusler alloy layer in two roles: as a
weak ferromagnet on the place of the F2 layer and as a half-metal on the place
of the F1 layer. In the first case, we obtained a large ordinary
superconducting spin-valve effect $\Delta T_c$ assisted by the triplet
superconducting spin-valve effect $\Delta T_c^{trip}$. In the second case, we
observed a giant magnitude of $\Delta T_c^{trip}$ reaching 0.5 K. An underlying
theory based on the solution of the Usadel equations using Kupriyanov-Lukichev
boundary conditions with arbitrary material parameters for all layers and
arbitrary boundary parameters for all interfaces is presented in Appendix. We
find a good agreement between our experimental data and theoretical results.
