With the rapid development of electronic and communication technology for military radars, the demand for microwave-absorbing materials in the low-frequency range with thin layers is growing. In this study, flexible Co3O4/CC (carbon cloth) composites derived from Co-MOFs (metal–organic frameworks) and CC are prepared using hydrothermal and thermal treatment processes. The flexible precursors of the Co-MOFs/CC samples are calcined with different calcination temperatures, for which the material structure, dielectric properties, and microwave absorption performance are changed. With the increases in calcination temperature, the minimum reflection loss of the corresponding Co3O4/CC composites gradually moves to the lower frequency with a thinner thickness. In addition, the Co3O4/CC composites with the 25 wt% filler loading ratio exhibit the minimum reflection loss (RL) of −46.59 dB at 6.24 GHz with a 4.2 mm thickness. When the thickness is 3.70 mm, the effective absorption bandwidth is 3.04 GHz from 5.84 to 8.88 GHz. This study not only proves that the Co3O4/CC composite is an outstanding microwave-absorbing material with better flexibility but also provides useful inspiration for research on wideband microwave absorption materials below 10 GHz.