It is commonly adopted that the widely used method of terahertz generation by tilted-pulse-front pumping of a lithium niobate crystal is optimized when both the tilted front of the pump optical pulse and the image of the diffraction grating, which introduces the tilt, are parallel to the exit surface of the crystal. We consider more general situations. In particular, we show that a deviation of the grating image plane from parallel to the crystal exit surface leads to deterioration of the terahertz beam quality but does not significantly affect the terahertz energy. If the grating image plane and the crystal exit surface are mutual parallel, but they are nonparallel to the pulse front, the terahertz beam quality remains unaffected and the terahertz energy can be unexpectedly higher (about two times for pump pulses of a ~400 fs duration) than in the standard scheme. We propose a modified design of the tilted-pulse-front scheme based on our findings.