We design a new version of a cycler orbit between Earth and Mars (known as the Aldrin cycler) in which we use low thrust to reduce the encounter velocities. We slow that by reducing the encounter velocities at both planets, the propellant needed by the taxis to perform hyperbolic rendezvous can be significantly reduced. If the V-infinity reduction is large enough, two-stage taxis can be used instead of three-stage taxis, thus further reducing the required injected mass to a low-Earth orbit (IMLEO) for a particular cycler trajectory. However, as the V-infinity decreases, the propellant expenditure for the cycler vehicle increases. Our trade studies (over seven synodic periods) show that V-infinity reductions can be effective in reducing the total IMLEO (i.e. cycler plus taxi) propellant for low-thrust Aldrin cycler missions.