Low-temperature ($T$) thermodynamic properties in magnetic fields ($B$) of the quasi-Kagome antiferromagnet CePdAl ($T_{rm N}sim$ 2.7 K) were studied by means of 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 low-field phase I, the existence of the partial Kondo screening state on the one-third Ce sites is supported from the thermodynamic point of view: (i) an appearance of the paramagnetic moment with one-third of the full moment around 3 T, and (ii) an occurrence of the entropy release $sim$0.3$R$ln2 around the observed crossover. On the other hand, a strong enhancement of $C/T$ is found around the point, where $T_{rm N}(B)$ meets the boundary of phase III, indicating the possible presence of a tri-critical point. Furthermore, it is suggested that the three metamagnetic transitions at $B_{{rm m}i}$ ($i=$1,2,3) come from successive increments of the Ising-type magnetic moment along the $c$-axis. In high-field state above 20 T, the behavior of $f$ electron can be rather described by the crystal-electric-field model.
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