The iterative ensemble Kalman filter (IEnKF) is widely used in inverse problems to estimate system parameters from limited observations. However, the IEnKF, when applied to nonlinear systems, can be plagued by poor convergence. Here we provide a comprehensive convergence analysis of the IEnKF and propose a new method to improve its convergence. A theoretical analysis of the standard IEnKF is presented and we demonstrate that the interaction between the nonlinearity of the forward model and the diminishing effect of the Kalman updates results in "early stopping" of the IEnKF, i.e. the Kalman operator converges to zero before the innovation is minimized. The steady state behavior of the early stopping phenomenon and its relation to observation uncertainty is demonstrate. We then propose an approach to prevent the early stopping by perturbing the covariance with hidden parameter ensemble resampling. The ensemble mean and covariance are kept unchanged during the resampling process, which ensures the Kalman operator at each iteration maintains a correct update direction. We briefly discuss the influence higher moments, such as kurtosis, of the resampling distribution on algorithm convergence. Parallel to the above developments, an example problem is presented to demonstrate the early stopping effect, and the application and merit of the proposed resampling scheme.