In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many S-integral points on ℙ1ℤ∖{0,1,∞}. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of S increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where S has size 2 which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.