When you say R⊆S, does that mean S has more equivalence classes than R (on A)?
No, I assume R ⊆ S means that R is a subset of S where both R and S are viewed as subsets of A x A. That is, S is coarser than R and thus has fewer equivalence classes than R.
The number of such S equals the number of equivalence relations on 4 elements. It is probably easier to calculate this number manually than to apply a general formula.