We show that the peculiarities of the electron band structure strongly affect the spin-valve effect in heterostructures consisting of a superconductor (S) and two ferromagnetic layers ${\mathrm{F}}_{1}$ and ${\mathrm{F}}_{2}$. For the $\mathrm{S}/{\mathrm{F}}_{1}/{\mathrm{F}}_{2}$ systems the energy shift between the electron bands in the S and ${\mathrm{F}}_{2}$ layers determines whether the critical temperature ${T}_{c}$ of the superconductor increases or decreases as a function of the angle $\ensuremath{\theta}$ between the magnetic moments in the ferromagnets. In the case of the half-metallic ${\mathrm{F}}_{2}$ layer the type of the spin-valve effect becomes dependent on the position of the only occupied spin band and the minimum of ${T}_{c}$ may correspond to the antiparallel, parallel, or noncollinear orientations of magnetic moments. Also, for the first time, we analytically demonstrate the possibility of the triplet spin-valve effect in the F/S/half-metal structures.