Herstein proved that $S$ can be enlarged to the set of all $a^2 p(a) - a$ with $p$ a polynomial (with integer coefficients).

EDIT.  Herstein's set may be maximal.  The set can't contain any polynomials whose vanishing would be consistent with the ring containing (nonzero) nilpotent elements, so nothing in $S$ can be divisible by $a^2$. The lower degree terms are also highly constrained by the condition that if there is $p$-torsion then no $p^2$-torsion.

EDIT-2.  Again, based on my memory of the $x^n=x$ problem, the point was to have conditions that prevent one from replacing $R$ with something like $R \oplus tQ \oplus uQ \oplus tuQ$ with $Q$ noncommutative and $u^2=ut=tu=t^2 = 0$, or adding a noncommutative torsion component, $R \oplus Q$ with $mQ=0$ for some $m > 1$.  I think that generating enough examples to show that Herstein's set is maximal should be easier than showing that Herstein's conditions imply commutativity.