In this paper, a new method for efficient generation of high-energy ion bunches via laser-induced cavity pressure acceleration (LICPA) is examined using one-dimensional particle-in-cell code PIC1D. It is found that for high laser beam intensities of the order of 1022 W/cm2 and for circular light polarization, a substantial increase in parameters of the accelerated ions is obtained when the target is placed inside a special cavity, into which the laser beam is introduced by a small hole. As compared to the pure radiation pressure acceleration scheme, the LICPA scheme leads to an increase in ion energies and the laser-to-ions energy conversion efficiency while the width of the ion energy spectrum are similar for both the schemes. Such a tendency was observed for all carbon targets (from 2 µm to 0.2 µm thick) investigated in the paper. The results of PIC1D simulations agree very well with predictions of the suitably generalized light sail model.