The increasing penetration of renewable energy necessitates efficient and site-independent energy storage solutions to ensure grid stability. Liquid air energy storage (LAES) has emerged as a promising candidate owing to its high energy density and absence of geographical constraints, while its round-trip efficiency remains limited by the energy-intensive air liquefaction process. This study proposes novel air liquefaction cycles integrated with vapor-injection refrigeration (VIRS) as an inter-stage precooling strategy to enhance the efficiency and compactness of microgrid-scale LAES systems. A comprehensive parametric analysis using Aspen HYSYS evaluates the integration of VIRS with Linde-Hampson, Claude, and Kapitza cycles, examining key variables including air compression pressure (pcomp,air), VIRS evaporation temperature (Tevap) and expander inlet temperature (Texp,in). Results demonstrate that VIRS precooling substantially enhances liquefaction performance across all cycles. The integration of VIRS precooling with Linde-Hampson, Claude and Kapitza cycles achieves the optimal SEC of 1.19 kWh/kg, 0.493 kWh/kg and 0.461 kWh/kg with the reduction ratios of 54.62%, 16.19% and 15.03% compared with the conventional air liquefaction cycles without precooling, while the counteractions among the parameters of VIRS and air process are comprehensively evaluated. Advanced exergy analysis quantifies the thermodynamic irreversibility and confirms the crucialness of air expander for mitigating exergy waste, while the precooling energy input contributes to substantial exergy savings within the air liquefaction process. These findings establish a foundation for developing distributed, miniaturized LAES systems with enhanced thermodynamic and economic viability for microgrid applications.