Production of high dynamic pressure using a strong shock wave is a topic of high relevance for high-energy-density physics, inertial confinement fusion, and materials science. Although the pressures in the multi-Mbar range can be produced by the shocks generated with a large variety of methods, the higher pressures, in the sub-Gbar or Gbar range, are achievable only with nuclear explosions or laser-driven shocks. However, the laser-to-shock energy conversion efficiency in the laser-based methods currently applied is low and, as a result, multi-kJ multi-beam lasers have to be used to produce such extremely high pressures. In this paper, the generation of high-pressure shocks in the newly proposed collider in which the projectile impacting a solid target is driven by the laser-induced cavity pressure acceleration (LICPA) mechanism is investigated using two-dimensional hydrodynamic simulations. A special attention is paid to the dependence of shock parameters and the laser-to-shock energy conversion efficiency on the impacted target material and the laser driver energy. It has been found that both in case of low-density and high-density solid targets, the shock pressures in the sub-Gbar range can be produced in the LICPA-based collider with the laser energy of only a few hundreds of joules, and the laser-to-shock energy conversion efficiency can reach values of 10–20%, by an order of magnitude higher than the conversion efficiencies achieved with other laser-based methods used so far.