Compressed-air energy storage (CAES) is an energy-storage option with a history of almost half a century. The concept of CAES is formed around the integration of air storage with a gas-fired power generator. Here, we introduce a methodology that a gas power-generating plant installs both air and natural-gas-storage systems to utilize the stored energy as well as the real economic value of natural gas, following the market dynamics. The energy-storage hybridization increases the operation complexity substantially. We present a detailed mixed-integer techno-economic formulation for operation scheduling of such a system. An example is also provided for a 180 MVV gas generator in Australia with results showing how the storage facilities could improve the revenues of the plant. The analyses also show the existence of optimal conditions for a mix of natural gas and storage sizes in a given regional market jurisdiction in order to achieve the highest economic revenue.