Linear inverse problems are very common in signal and image processing. Algorithms that solve such problems typically involve several unknown parameters that need to be tuned. Here we consider an iterated shrinkage method that is based on the separable surrogate functions (SSF) idea, which exploits the sparsity of the unknown vector in an appropriate representation. The key parameter controlling the algorithm's success is the prior weight, denoted λ. Previous work has addressed the automatic tuning of λ based on a generalized Stein Unbiased Risk Estimator (SURE) of the mean-squared error (MSE). The approach taken was to obtain a constant value of λ that leads to optimized results over a given set of iterations. In this work we also rely on the generalized SURE, and propose an alternative, and highly effective method for tuning λ. Our algorithm chooses λ per iteration, based on the local estimated risk, considering the current iteration and a possible short look-ahead. We demonstrate this method and its superiority over the global approach both in terms of the resulting MSE and the convergence rate. We also show that the proposed scheme serves as a very reliable automatic halting mechanism for the iterative process.