We will discuss the existence of rational (multi)sections and unirulings for projective families f:X→P1 with at most two singular fibres. Specifically, we will discuss two ingredients for constructing the above rational curves. The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains. The second is a degeneration to the normal cone argument that allows one to produce closed curves in X from open curves (which are produced using local symplectic cohomology) in the complement of X by a singular fibre.