The integration of stiff ordinary differential equation (ODE) systems associated with detailed chemical kinetics is computationally demanding in practical combustion simulations. Despite the various approaches in expediting the computational efficiency, it is still necessary to optimise the cell-wise calculation in operator-splitting type simulations of reactive flow. In this work, we proposed an improved stiff-ODE solver framework targeting to speed up the simulation of reactive flow in OpenFOAM. This framework combines the Radau-IIA and backward differentiation formula (BDF) ODE-integration algorithms, the pyJac-based fully analytical Jacobian formulation, and dense-based LAPACK and sparse-based KLU sophisticated linear system solvers. We evaluate the performance of the efficient solver framework on various benchmark combustion problems across a wide range of chemical kinetic complexities. A comprehensive investigation of the key elements of stiff ODE solvers is conducted in the homogeneous reactor, focusing respectively on the influences of error tolerance, integration time interval, Jacobian evaluation methodology, and linear system solver on the accuracy and efficiency trade-off. More realistic simulation results are presented regarding the one-dimensional laminar flame and three-dimensional turbulent flame. The results indicate that the Radau-IIA is more preferable in both efficiency and accuracy compared with the widely used BDF and Seulex methods for large integration interval, whereas the differences between three methods diminish as the integration time interval decreases. In all cases, it is found that the full analytical Jacobian is more advantageous for small mechanisms of species number around 50-100 while the approximated formulation of Jacobian is recommended for larger ones. Furthermore, the more robust linear system solvers provide significant improvement on computational efficiency with the dense-based LAPACK solver being more suitable for small to moderate-scale mechanisms while sparse-based KLU being superior for large-scale mechanisms. The proposed efficient solver framework in its optimal configuration obtains more than 2.6 times speedup in realistic high-fidelity flame simulation with a 57 species combustion mechanism.