$M$-matrix is applied widely in mathematical physics, cybernetics, electric system and so on. In recent years, the upper bound estimate on the infinity norm of inverse matrix of a strictly diagonally dominant $M$-matrix has become an important problem. Firstly, some new notations are introduced in this paper. Then, applying the range for the elements of inverse matrix and algebra methods, some new upper bounds for the infinity norm of inverse matrix of a strictly diagonally dominant $M$-matrix are obtained. Finally, the theory analysis and numerical results show that the new upper bound estimates improve some of the related results.