Smoothed particle hydrodynamics (SPH) method as one of the meshless Lagrangian methods has been widely used to simulate problems with free surface. The traditional SPH method suffers from so-called tensile instability, which may eventually result in numerical instability or complete blowup during the simulation of bubble/droplet dynamics. A new pressure-correction equation is proposed to efficiently transport the local pressure to the neighboring area during the impact of incompressible/compressible fluid and reduce the disorder of particle distribution. Consequently, the accuracy and the efficiency of the SPH method can be dramatically improved. New treatments to the surface tension and solidification are also proposed to manipulate SPH particles near the free surface and the solidification interface. The improved SPH method has been used to simulate droplet impact, spreading, and solidification. It is evident that the new method can handle the droplet contraction problem without causing numerical instability. The numerically predicted flattening ratio of the splat due to droplet impact is in good agreement with the analytical prediction. The results demonstrate that the improved SPH model is a powerful tool to study droplet spreading and solidification.