Utilities today are in need of tools and techniques that will enable them to predict the dynamic stability and reliability of the grid in the real-time. The problem is challenging because of the large number of contingencies that are to be simulated. In this paper a fast method for power system contingency screening and ranking for small signal stability assessment is presented which essentially reduces the number of contingencies for detailed evaluation. The proposed method avoids repeated computation of eigenvalues for all possible post-contingency scenarios. Instead, the eigenvalues corresponding to critical modes for post-outage conditions are estimated based on first-order eigenvalue-sensitivity using just the nominal condition eigenvalues and post outage system state matrices. Since a critical outage condition can produce a large change in the eigenvalues, the first order prediction might not have acceptable accuracy. To overcome this issue, a second-order correction is applied which needs the computation of the eigenvectors corresponding to the eigenvalues of interest. A significant reduction in computation time for evaluation of post contingency eigenvalues using the proposed method is demonstrated. A case study on a 16-machine NETS-NYPS system shows promising results.