We propose a fast method for solving compressed sensing, Lasso regression, and Logistic Lasso regression problems that iteratively runs an appropriate solver using an active set approach. We design a strategy to update the active set that achieves a large speedup over a single call of several solvers, including gradient projection for sparse reconstruction (GPSR), lassoglm of Matlab, and glmnet. For compressed sensing, the hybrid of our method and GPSR is 31.41 times faster than GPSR on average for Gaussian ensembles and 25.64 faster on average for binary ensembles. For Lasso regression, the hybrid of our method and GPSR achieves a 30.67-fold average speedup in our experiments. In our experiments on Logistic Lasso regression, the hybrid of our method and lassoglm gives an 11.95-fold average speedup, and the hybrid of our method and glmnet gives a 1.40-fold average speedup.