Low-temperature (T) thermodynamic properties of the quasi-Kagome antiferromagnet CePdAl (TN ∼ 2.7 K) in magnetic fields (B) were studied by magnetization and specific-heat C(T, B) measurements. The unusual magnetic phase diagram, including three thermodynamic phases I-III and crossover anomalies, has been obtained. In the low-field phase I, the existence of the partial Kondo screening state on one-third of the Ce sites is supported from the following thermodynamic view points: (i) the appearance of a paramagnetic moment with one-third of the full moment around 3 T and (ii) the occurrence of an entropy release of ∼0.3R ln 2 around the observed crossover. On the other hand, a strong enhancement of C=T was found around the point where TN(B) meets the boundary of phase III, indicating the possible presence of a tricritical point. Furthermore, it is suggested that the three metamagnetic transitions at Bmi (i = 1, 2, 3) originate from successive increments of the Ising-type magnetic moment along the c-axis. In the high-field region above 20 T, the behavior of f electrons can be roughly described by the crystal-electric-field model.