We demonstrate electrically tunable twisted nematic (TN) aligned liquid crystal (LC) terahertz phase shifters with a novel structure, in which a single LC composite polymer thin film is inserted in the middle of the LC cell. For designing such types of phase shifters, we have applied the Mauguin formalism used in LC-based switches for applications in the visible. Experimental results are in good agreement with those predicted by the theoretical model. Using this design, the phase shift has been increased by 14.3° at 1.2 THz in comparison to the conventional design (98.1°). The threshold voltage has been lowered from 0.81 VRMS to 0.48 VRMS while the driving voltage for quarter-wave (π/2) phase shift operation has been reduced from 8.4 VRMS to 6.3 VRMS at 1.2 THz. The dynamic response of devices also enhances significantly. Such design has also successfully been adopted to demonstrate a 2π phase shifter with similar improvement. Full-wave operation is achieved at a bias of 25 VRMS at 1.2 THz using a highly birefringent LC material. Order parameter calculations show that the polymer film improves significantly LC molecular alignment in the thick LC cell for THz applications. We have also compared the figure of merits (FOMs) of the present devices with previously reported THz π/2 and 2π phase shifters.