We present and analyse a backtracking strategy for a general Fast Iterative Shrink-age/Thresholding Algorithm which has been recently proposed in [11] for strongly convex objective functions. Differently from classical Armijo-type line searching, our backtracking rule allows for local increase and decrease of the Lipschitz constant estimate along the iterations. For such strategy accelerated convergence rates are proved and numerical results are shown for some exemplar imaging problems.