In this paper, we extend a methodology developed recently to study type-III intermittency considering different values of the noise intensity and the lower boundary of reinjection (LBR). We obtain accurate analytic expressions for the reinjection probability density (RPD). The proposed RPD has a piecewise definition depending on the nonlinear behavior, the LBR value, and the noise intensity. The new RPD is a sum of exponential functions with exponent α + 2, where α is the exponent of the noiseless RPD. The theoretical results are verified with the numerical simulations.