This paper refers to Clarke generalized gradient for a smooth composition ofmax-type functions of the form: f(x) = g(x, maxjej1 f1j(x),..., maxjeJm fmj(x) ),where x∈Rn, Ji, i = 1,.,m are finite index sets, g and fij,j ~ Ji, i = 1,. , m,are continuously differentiable on Rm+n and Rn, respectively. In a previous paper,we proposed an algorithm of finding an element of Clarke generalized gradient forf, at a point. In that paper, finding an element of Clarke generalized gradient for f,at a point, is implemented by determining the compatibilities of systems of linearinequalities many times. So its computational amount is very expensive. In thispaper, we will modify the algorithm to reduce the times that the compatibilitiesof systems of linear inequalities have to be determined.