Electrical conductivity directly correlates with tissue functional information such as blood and water contents, and quantitative extraction of tissue conductivity is of significant importance for disease detection and diagnosis using microwave-induced thermoacoustic tomography (TAT). The existing quantitative TAT (qTAT) approaches capable of extracting tissue conductivity require two steps for the recovery of conductivity. Such two steps approaches depend on an accurate knowledge of the microwave energy loss distribution in tissue and offer a slow computational convergence rate. The purpose of this study is to develop a new algorithm to reconstruct tissue conductivity with higher reconstruction accuracy and greater computational efficiency. We propose an improved qTAT method for direct recovery of tissue conductivity from thermoacoustic data measured along the boundary with only one step without the dependence of microwave energy loss information. The feasibility of our one-step qTAT method is validated in both simulated and tissue-mimicking phantom experiments with single-target and multi-target configurations with different contrast levels. Compared with the previous two-step methods, our one-step qTAT method improves the accuracy of conductivity recovery with approximately one-fold reduction in the mean absolute error (MAE) and root mean square error (RMSE) with p-values greater than 0.05. In addition, the convergence rate is improved by more than two folds for the one-step method. The study demonstrates that new method can quantitatively reconstruct conductivity of tissue more accurately and efficiently over the existing qTAT methods, leading to potentially enhanced accuracy for disease detection and diagnosis.