IGR J11014-6103 (also known as the Lighthouse Nebula) is composed of a bow-shock pulsar wind nebula (PWN) and large-scale X-ray jet-like features, all powered by PSR J1101-6101. Previous observations suggest that the jet features stem from a ballistic jet of relativistic particles. In order to confirm the nature of the jet and the counter-jet, we obtained a new deep 250 ks Chandra observation of the Lighthouse Nebula. We performed detailed spatial and spectral analysis of all X-ray components of the system. The X-ray PWN is now better resolved and shows a peculiar morphology resembling the shape of an arrow. The overall helical pattern of the main jet is confirmed. However, there are large deviations from a simple helical model at small and large scales. Significant extended emission is now detected, encompassing the main jet all along its length. The presence of an apparent gap along the main jet at ~50″ distance from the pulsar is confirmed; however, the surrounding extended emission prevents conclusions on the coherence at this position of the jet. The counter-jet is now detected at high statistical significance. In addition, we found two small-scale arcs departing from the pulsar towards the jets. We also looked for possible bow-shock emission due to the pulsar motion, with a short VLT/FORS2 H-α observation. No clear emission is found, most likely because of the contamination from a diffuse nebulosity. The results of our X-ray analysis show that both a ballistic jet scenario and an alternative scenario involving the diffusion of particles along pre-existing interstellar magnetic field lines are able to satisfactorily explain some of the observational evidence, but cannot fully reproduce the observations.