129Xe NMR spectroscopy of polymers can provide important information on void spaces, sometimes called free volume, in polymers. Unfortunately, the spectroscopy's low sensitivity has limited its widespread use in both academic and industrial research. In order to overcome such a difficult situation, hyper-CEST method which employs hyperpolarization and CEST techniques, is examined after the introduction of recirculation and subtraction modes. Alongside the incorporated stopped-flow technique, these modes were very efficient in detecting very weak hidden signals from cellulose nanofiber (CNF) and silk fibroin (SF) films and in discussing the void space in these polymers. From the analysis of detailed saturation frequency dependence in the increment of 100Hz, the chemical shifts of hidden peaks were successfully determined to give reasonable values for the size of void space in CNF and SF. Application on thermoplastic polyurethane film also supported our method of analysis. The subtraction mode was very efficient in judging the presence or absence of any peak at a fixed saturation frequency. These facts support that the mode will surely be useful in the future exploratory study of very weak hidden signals.