Efficiently exploiting parallelism remains a challenging problem in multicore processors. For many algorithms, executing tasks in some priority order results in a work efficient execution. However, searching high-priority tasks requires communication that hampers performance. A concurrent priority scheduler (CPS) selects high-priority tasks and schedules them on different cores. Modern CPS designs offer various strategies to select high-priority tasks at low communication costs for improved performance. However, they do not explicitly track the priority of tasks, and cannot adjust task distribution if low priority tasks are being processed by the cores. Moreover, they cannot estimate the right amount of communication required to select high-priority tasks. This paper makes a critical observation that the cores’ priority drift can be quantified and used for better performance. A novel CPS design is proposed that uses priority as a signal to co-optimize drift and communication at runtime. Furthermore, compute intensive task transfer and processing aspects of the CPS are offloaded to per-core local hardware at low cost to enhance performance. HD-CPS is shown to consistently improve performance over several state-of-the-art software-based and hardware-assisted CPS designs. With hardware-assist, it approaches near linear performance scaling as a function of core counts for large scale shared memory multicores.