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In the present work, we study the combined <i>m</i>-consecutive-<i>k</i>-out-of-<i>n</i>: <i>F</i> and <i>k_c</i>-out-of-<i>n</i>: <i>F</i> reliability systems, which consist of independent and identically distributed components. Two different redundancy policies are considered, and their general frameworks are described and illustrated. The main objective of the paper refers to the investigation of the effect of adding cold standby redundancy to the system at the the system level and the component level. Exact formulae for determining the crucial characteristics of the enhanced structure, such as its survival function, the mean time to failure and the mean residual lifetime, are provided. All formulae proved in the present manuscript are explicit expressions which are based on the signature vector of the resulting reliability schemes. An extensive numerical investigation is carried out to shed light on the performance of the combined <i>m</i>-consecutive-<i>k</i>-out-of-<i>n</i>: <i>F</i> and consecutive <i>k_c</i>-out-of-<i>n</i>: <i>F</i> reliability systems with cold standby redundancy. Some concluding remarks and comments are provided upon the determination of the optimal design parameters.