The Princeton Field-Reversed Configuration (PFRC) experiment employs an odd-parity rotating-magnetic-field (RMFo) current drive and plasma heating system to form and sustain high- plasmas. For radial confinement, an array of coaxial, internal, passive flux conserver (FC) rings applies magnetic pressure to the plasma while still allowing radio-frequency RMFo from external coils to reach the plasma. The 3-ms pulse duration of the present experiment is limited by the skin time fc of its room-temperature copper FC rings. To explore plasma phenomena with longer characteristic times, the pulse duration of the next-generation PFRC-2 device will exceed 100 ms, necessitating FC rings with fc > 300 ms. In this paper we review the physics of internal, discrete, passive FCs and describe the evolution of the PFRC's FC array. We then detail new experiments that have produced higher-performance FC rings that contain embedded high-temperature superconducting (HTS) tapes. Several HTS tape winding configurations have been studied, and a wide range of extended skin times, from 0.4 s to >103 s, has been achieved. The new FC rings must carry up to 2.5 kA of current to balance the expected PFRC-2 plasma pressure, so the dependence of the HTS-FC critical current on the winding configuration and operating temperature has also been studied. From these experiments, the key HTS-FC design considerations have been identified and HTS-FC rings with the desired performance characteristics have been produced.