In this paper we propose QContext, a new compiler structure that incorporates context-aware and topology-aware decompositions. Because of circuit equivalence rules and resynthesis, variants of a gate-decomposition template may exist. QContext exploits the circuit information and the hardware topology to select the gate variant that increases circuit optimization opportunities. We study the basis-gate-level context-aware decomposition for Toffoli gates and native-gate-level context-aware decomposition for CNOT gates. Our experiments show that QContext reduces the number of gates as compared with the state-of-the-art approach, Orchestrated Trios.