Personal thermal management materials integrated with phase-change materials have significant potential to satisfy human thermal comfort needs and save energy through the efficient storage and utilization of thermal energy. However, conventional organic phase-change materials in a solid state suffer from rigidity, low thermal conductivity, and leakage, making their application challenging. In this work, polyethylene glycol (PEG) was chosen as the phase-change material to provide the energy storage density, polyethylene oxide (PEO) was chosen to provide the backbone structure of the three-dimensional polymer network and cross-linked with the PEG to provide flexibility, and carbon nanotubes (CNTs) were used to improve the mechanical and thermal conductivity of the material. The thermal conductivity of the composite fiber membranes was boosted by 77.1% when CNTs were added at 4 wt%. Water-resistant modification of the composite fiber membranes was successfully performed using glutaraldehyde-saturated steam. The resulting composite fiber membranes had a reasonable range of phase transition temperatures, and the CC4PCF-55 membranes had melting and freezing latent heats of 66.71 J/g and 64.74 J/g, respectively. The results of this study prove that the green CC4PCF-55 composite fiber membranes have excellent flexibility, with good thermal energy storage capacity and thermal conductivity and, therefore, high potential in the field of flexible wearable thermal management textiles.