Lower-limb exoskeletons (LLE) have proven to be effective tools in rehabilitation for diminishing gait impairments. However, modern LLE control systems based on the assistance-as needed concept face challenges in providing appropriate interaction torques to the user. This is especially true when it is necessary to make the exoskeleton transparent, i.e. with zero user robot interaction torques (τUR), even during walking speed variations, when no assistance is required. To address these limitations, in this work we designed a novel transparent control system that utilizes a baseline torque controller, which generates a torque profile based on τUR from previous strides, and drives the actuators, leading to a lower τUR, along with a synchronization layer comprising a pool of adaptive oscillators (AOs) to estimate the user's gait phase and adapt the controller to different walking speeds. Additionally, a zero-torque controller was used in parallel to the main torque controller to enhance users' volitional control and balance. The proposed controller was experimentally tested on a six-DoF LLE with eight healthy participants who walked for 200 gait strides at three walking speeds. The baseline torque controller was shown to adapt to different walking speeds and successfully improve the exoskeleton transparency in terms of user-robot interaction, reducing the average τUR by 40% of the entire exoskeleton and 70% of the hip joint at 0.8 m/s, and gait kinematics compared to our prior controller based on a zero impedance model.