Target design study to improve the quality of an accelerated proton beam from the interaction of a high-intensity short pulse laser with an overdense plasma slab has been accomplished by using a two-dimensional, fully electromagnetic and relativistic particle-in-cell (PIC) simulation. The target consists of a thin core part and a thick peripheral part of equivalent plasma densities, while the ratio of the radius of the core part to the laser spot size, and the position of the peripheral part relative to the fixed core part were varied. The positive effects of this core-peripheral target structure could be expected from the knowledge of the typical target normal sheath acceleration (TNSA) mechanism in a laser-plasma interaction, and were apparently evidenced from the comparison with the case of a conventional simple planar target and the case of the transversal size reduction of the simple planar target. Improvements of the beam qualities including the collimation, the forward directionality, and the beam divergence were verified by detailed analysis of relativistic momentum, angular directionality, and the spatial density map of the accelerated protons.