Studies of entanglement between the neutron's degrees of freedom (DOF) have successfully confirmed the violation of a Bell-like inequality, Kochen-Specker-like phenomena, and the characteristics of tripartite Greenberger-Horne-Zeilinger (GHZ) state. Here, quantum contextuality, rather than quantum non-locality, is the basis of peculiarities of double- and triple-entangled quantum states. While so-called quantum state tomography has been established for spin-path entanglement, simpler schemes to distinguish more general entanglements are still subject to hot theoretical debate. Recently, a new scheme has been proposed by theoreticians in Vienna, who have close contact with us. We found that this scheme is very suitable to experimentally characterize entanglement in a single-neutron system. In parallel, we developed new methods to efficiently entangle the spin, path, and energy DOF, e.g., most recently RF spin flippers with independently adjustable energy shifts for the interferometer. With this new progress, we are going to continue our experiments concerning the newly proposed scheme of entanglement witness and further examine entanglements between the neutron's DOF.