We have enhanced the tunability of a terahertz-wave parametric generator using MgO:LiNbO3 pumped by a subnanosecond, passively Q-switched, microchip Nd:YAG laser. This pump source allows high-intensity pumping without damage to the nonlinear crystal and generates a narrow linewidth and tunable terahertz wave with injection seeding by an external cavity diode laser for the idler wave. The high-intensity pumping causes a gain curve broadening of the terahertz-wave parametric generation, especially in the high-frequency region. We obtained an output terahertz wave with a tuning range of 0.9-3 THz (100-330 microm), a peak power of 100 mW, and a linewidth of less than 10 GHz. This room temperature operated, palm-sized source promises to be a widely tunable terahertz-wave source that is suited to a variety of applications.