IN OUR WORK WE CONSIDER THE FOLLOWING PROBLEM IN THE CONTEXT OF TELEPORTATION: AN UNKNOWN PURE STATE HAS TO BE TELEPORTED AND THERE ARE TWO LABORATORIES WHICH CAN PERFORM THE TASK. ONE LABORATORY USES A PURE NON-MAXIMALLY ENTANGLED CHANNEL BUT HAS A CAPABILITY OF PERFORMING THE JOINT MEASUREMENT ON BASES WITH A CONSTRAINED DEGREE OF ENTANGLEMENT; THE OTHER LAB MAKES USE OF A MIXED X-STATE CHANNEL BUT CAN PERFORM A JOINT MEASUREMENT ON BASES WITH HIGHER ENTANGLEMENT DEGREES. WE COMPARE THE AVERAGE TELEPORTATION FIDELITY ACHIEVED IN BOTH CASES, FINDING THAT THE FIDELITY ACHIEVED WITH THE X-STATE CAN SURPASS THE OBTAINED WITH A PURE CHANNEL, EVEN THOUGH THE X-STATE IS LESS ENTANGLED THAN THE LATTER. WE FIND THE CONDITIONS UNDER WHICH THIS EFFECT OCCURS. OUR RESULTS EVIDENCE THAT THE ENTANGLEMENT OF THE JOINT MEASUREMENT PLAYS A ROLE AS IMPORTANT AS THE ENTANGLEMENT OF THE CHANNEL IN ORDER TO OPTIMIZE THE TELEPORTATION PROCESS. WE INCLUDE AN EXAMPLE SHOWING THAT THE AVERAGE FIDELITY OF TELEPORTATION OBTAINED WITH A WERNER STATE CHANNEL CAN BE GREATER THAN THAT OBTAINED WITH A BELL STATE CHANNEL.