Let $G$ be a primitive permutation group acting on a finite set $X$. The orbital diameter $\mathrm{diam}(X,G)$ is defined to be the supremum of the diameters of the (connected) orbital graphs of $G$ after disregarding the directions of all edges in the graphs. This invariant is studied in the case when $G$ is an almost simple group in a standard action. A lower bound is given for $\mathrm{diam}(X,G)$ and we provide a partial classification of pairs $(X,G)$ for which the orbital diameter is at most $2$.