Solving acoustic problems in engineering usually resort to numerical methods. The conventional finite element method (FEM) and the boundary element method (BEM) are widely used for complex acoustic problems. But the conventional FEM requires much time to acquire satisfactory meshes for the accuracy of solutions, and thus is restricted to acoustic problems with low and middle frequencies. More methods are proposed to solve the problems mentioned above. In this work, a coupled overlapping finite element method is employed by combining the overlapping finite element method (OFEM) and the modified Dirichlet-to-Neumann (mDtN) boundary condition to solve underwater acoustic scattering problems. The main difference between the OFEM and the conventional FEM lies in the construction of the local field approximation. In the OFEM, virtual nodes are employed to form the partition of unity functions in the local field while no degree of freedom is assigned to these virtual nodes, which makes the linear dependence issue in other generalized finite element method suppressed. Moreover, the user-defined enrichment functions can be flexibly utilized in the local field, and thus the numerical dispersions can be significantly mitigated. To truncate the infinite problem domain and satisfy the Sommerfeld radiation condition, an artificial boundary is constructed by incorporating the mDtN technique. Several numerical examples have been done with the OFEM and the conventional FEM. According to the results, the OFEM can significantly reduce the numerical error and improve the accuracy of solutions. And the OFEM is proved to be insensitive to distorted meshes, indicating that the proposed method is promising to predict underwater acoustic scattering problems.