We show that quasimonoenergetic proton beams can be generated through a multi-ion thick target irradiated by a circularly polarized laser pulse. After disrupted by the transverse instabilities in the laser pressure acceleration process, heavy ions as majority species can still provide a co-moving electric field. Different from the dynamics using ultrathin foil, protons with small doped rates can experience a full tri-stage quasimonoenergetic acceleration (hole boring, sheath boosting, and free expansion stages) in this scenario. A theoretical model is developed to explain the proton energy evolution in detail and verified by two-dimensional particle-in-cell simulations. The scaling of proton energy with laser intensity indicates that the 200 MeV proton beam with narrow energy spread (3%-10%) and sufficiently large charges (1010-1011) required for medical applications can be obtained using 100s TW class laser systems in near future.