  For a graph $G = (V, E)$ embedded in the projective plane, let
$\mathcal{F}(G)$ denote the set of faces of $G$. Then, $G$ is called a
$C_n$-face-magic projective graph if there exists a bijection $f: V(G) \to \{1,
2, \dots, |V(G)|\}$ such that for any $F \in \mathcal{F}(G)$ with $F \cong
C_n$, the sum of all the vertex labels around $C_n$ is a constant $S$. We
consider the $m \times n$ grid graph, denoted by $\mathcal{P}_{m,n}$, embedded
in the projective plane in the natural way. Let $m \geqslant 3$ and $n
\geqslant 3$ be odd integers. It is known that the $C_4$-face-magic value of a
$C_4$-face-magic labeling on $\mathcal{P}_{m,n}$ is either $2mn+1$, $2mn+2$, or
$2mn+3$. The characterization of $C_4$-face-magic labelings on
$\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+2$ is known. In this
paper, we determine a category of $C_4$-face-magic labelings on
$\mathcal{P}_{m,n}$ for which the $C_4$-face-magic value is either $2mn+1$ or
$2mn+3$. It is conjectured that these are the only $C_4$-face-magic labeling on
$\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+1$ or $2mn+3$.
