An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail.