This is a continuation of the paper (Comm. Math. Phys. 230 (2002) 329) on the study of the compressible Navier–Stokes equations for isentropic flow when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. This assumption comes from physical consideration and it also gives the well-posedness of the Cauchy problem. A new global existence result is established by some new a priori estimates so that the interval for the power of the density in the viscosity coefficient is enlarged to (0,-1/3)