Multi-segment continuum robots offer enhanced safety during surgery due to their inherent passive compliance. However, they suffer poor position tracking performance due to flexibility of their actuation lines, structural compliance, and actuation coupling effects between segments. The need for control methods addressing accurate tracking for multisegment continuum robots is magnified by increased precision requirements of surgical procedures employing these structures. To address this need, this paper proposes a tiered controller that uses both extrinsic and intrinsic sensory information for improved performance of multi-segment continuum robots. The higher tier of this controller uses configuration space feedback while the lower tier uses joint space feedback and a feed-forward term obtained with actuation compensation techniques. We prove the stability of this controller using Lyapunov's direct method and experimentally evaluate its performance on a three-segment multi-backbone continuum robot. Results demonstrate its efficacy in enhancing regulation and tracking performance. It is shown that the controller mitigates the effects of actuation coupling between robot's sub-segments and decreases phase lag. These results suggest that this tiered controller will enhance telemanipulation performance of multi-segment continuum robots.