The transverse folding algorithm [M. C. Bañuls et al., Phys. Rev. Lett. 102, 240603 (2009)] is a tensor network method to compute time-dependent local observables in out-of-equilibrium quantum spin chains that can overcome the limitations of matrix product states when entanglement grows slower in the time than in the space direction. We present a contraction strategy that makes use of the exact light cone structure of the tensor network representing the observables.