I think there is enough information in Chapter III of Ward's paper to determine the classes of 3-singular elements, and the paragraph numbers below refer to that).

Let $P \in {\rm Syl}_3(G)$. Then $|P|=q^3$, with upper and lower central series of $P$ both $1 < C < P_1 < P$ (using Ward's notation), where $|C|=q$, $P_1$ is elementary abelian of order $q^2$, and $P/P_1$ is elementary abelian.

The normalizer of $P$ is a semidirect product of $P$ with a cyclic group of order $q-1$, which acts fixed-point-freely on $C$ and on $P/P_1$, but acts as a cycle of order $(q-1)/2$ on the middle layer $P_1/P$. The element of order 2 in this cyclic group has centralizer $C_2 \times L_2(q)$.

The $q-1$ elements in $C \setminus \{1\}$ form a single class. The $q^2-q$ elements of $P_1 \setminus C$ (which have order 3) split into two classes (paragraph 7). These elements centralize an element of order 2, and there are also two classes of elements of order 6. Finally, the $q^3-q^2$ elements in $P \setminus P_1$ split into three classes of elements of order 9 (paragraphs 9,10,11).