As renewable electricity generation capacity increases, energy storage will be required at larger scales. Compressed Air Energy Storage (CAES) at large scales, with effective management of heat, is recognised to have potential to provide affordable grid-scale energy storage. Where suitable geologies are unavailable, compressed air could be stored in pressurised steel tanks above ground, but this would incur significant storage costs. Liquid Air Energy Storage (LAES), on the other hand, does not need a pressurised storage vessel, can be located almost anywhere, has a relatively large volumetric exergy density at ambient pressure, and has relatively low marginal cost of energy storage capacity even at modest scales. However, it has lower roundtrip efficiency than compressed air energy storage technologies. This paper carries out thermodynamic analyses for an energy storage installation comprising a compressed air component supplemented with a liquid air store, and additional machinery to transform between gaseous air at ambient temperature and high pressure, and liquid air at ambient pressure. A roundtrip efficiency of 42% is obtained for the conversion of compressed air at 50 bar to liquid air, and back. The proposed system is more economical than pure LAES and more economical than a pure CAES installation if the storage duration is sufficiently long and if the high-pressure air store cannot exploit some large-scale geological feature.