The transport of an infinitely thin, hard rod in a random, dense array of point obstacles is investigated by molecular dynamics simulations. Our model mimics the sterically hindered dynamics in dense needle liquids. Transport becomes increasingly fast at higher densities, and we observe a power-law divergence of the diffusion coefficient with exponent 0.8. This phenomenon is connected with a new divergent time scale, reflected in a zigzag motion of the needle, a two-step decay of the velocity-autocorrelation function, and a negative plateau in the non-Gaussian parameter. Finally, we provide a heuristic scaling argument for the new exponent.