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The Morris method is an effective sample-based sensitivity analysis technique that has been applied in various disciplines. To ensure a more proper coverage of the input space and better performance, an enhanced framework for Morris is proposed by considering the combination of a sequential sampling strategy and the traditional Morris method. The paper introduces utilizing progressive Latin hypercube sampling to generate starting points while progressively preserving Latin hypercube property. Then the calculations for Elementary Effects, which occupies the major computational cost of Morris, become sequential. An adaptive stop criterion is also constructed to end the algorithm when the convergence condition is satisfied. Therefore, the proposed procedure makes the cost of Morris more manageable and minimizes the computational burden by conducting only model runs that are necessary to achieve reliable results. Two numerical examples and two real-world cases are given to illustrate the effectiveness and robustness of the framework.