In vehicular platoons, disturbances can be amplified significantly downstream in the platoon if not adequately addressed, potentially leading to traffic jams or collisions. This paper tackles the challenge in maintaining string stability of heterogeneous vehicular platoons under a bidirectional communication topology and a refined constant time headway spacing policy. First, unlike the commonly used constant spacing policy and constant time headway policy, a refined constant time headway policy that combines the benefits of constant spacing and constant time headway policies is presented to enhance platooning safety while maintaining traffic efficiency. Second, a distributed adaptive estimator is designed such that each follower ensures its real-time estimation on the inaccessible leader's full states. The proposed estimators also well accommodate the heterogeneity among vehicles, allowing distinct inertial lag parameters of the vehicle longitudinal dynamics. Third, leveraging a bidirectional communication topology, a distributed scalable platoon controller is developed to guarantee the desired individual stability and string stability requirements of the vehicle platoon without the need of any global information of the communication topology. Formal sufficient conditions are provided on the existence of the desired estimator and controller gains. Finally, numerical simulations are conducted to verify the efficacy of the derived theoretical results. The simulations highlight the string stability and superiority of the proposed spacing policy, demonstrating the effectiveness of the proposed platooning control method.