Note on Iitaka Conjecture $C_{n,m}$

Let $f: X \to Y$ be an algebraic fiber space, where $X$ and $Y$ are smooth projective varieties of dimensions $n$ and $m$, respectively. In \cite{Caopaun}, Cao and P\u{a}un proved $C_{n,m}$ when $Y$ has maximal Albanese dimension. In this paper, we prove $C_{n,m}$ in the case where the Albanese dimension $\alpha(Y)$ of $Y$ satisfies $\alpha(Y) \geq m-2$.