<p>The present paper provides the study on the composition operators due to Szász-Mirakyan with the generalized Szász-Durrmeyer operators. We get an interesting new operator based on Touchard polynomials. Also, the new operator can be further decomposed into two different operators. This study is extension of the recent work, where we have discussed such compositions with reverse order. We find some convergence behavior of these composition operators in sense of point-wise estimates and quantitative estimates in terms of modulus of continuity. We also provide some graphs to have convergence visualization for different <i>n</i>.</p>