The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is regarded as the state-of-the-art among a number of proximal gradient-based methods used for addressing large-scale optimization problems with simple but non-differentiable objective functions. However, the efficiency of FISTA in a wide range of applications is hampered by a simple drawback in the line search scheme. The local estimate of the Lipschitz constant, the inverse of which gives the step size, can only increase while the algorithm is running. As a result, FISTA can slow down significantly if the initial estimate of the Lipschitz constant is excessively large or if the local Lipschitz constant decreases in the vicinity of the optimal point. We propose a new FISTA-like method endowed with a robust step size search procedure and demonstrate its effectiveness by means of a rigorous theoretical convergence analysis and simulations.