  Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We
introduce a type of modified Schmidt games which is induced by $f$ and played
on any unstable manifold. Utilizing it we generalize some results of \cite{Wu}
as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) :=
\{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in
M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that
$E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$,
which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$.
Then for any nonempty open set $V \subset M$ we show that $E(f, y) \cap V$ has
full Hausdorff dimension equal to $\dim M$, by using a technique of
constructing measures supported on $E(f, y)$ with lower pointwise dimension
approximating $\dim M$.
