Within the framework of two-dimensional linear elasticity, the unilateral frictionless contact between two conformal cylindrical surfaces is governed by an integral equation in which the first and second Dundurs' constants are involved. In the case of elastic similarity characterized by the zero second Dundurs' constant, the integral equation is considerably simplified so as to lend itself to a closed-form solution. However, in the case of elastic dissimilarity defined by the non-zero second Dundurs' constant, the question of obtaining a closed-form solution to the integral equation is a much tougher one. Starting from the integral equation established by To et al. [To, Q.D., He Q.-C., Cossavella, M., Morcant, K., Panait, A., 2007. Closed-form solution for the contact problem of reinforced pin-loaded joints used in glass structures. Int. J. Solids Struct. 44, 3887-3903] for the conformal contact problem originating from a reinforced pin-loaded joint used in tempered glass structures, the present work proposes a new approximate analytical method to solve it in the case of elastic dissim-ilarity by minimizing an error function. The derived closed-form solution, valid not only for the conformal contact between a pin and an infinite holed plate but also for the one between a pin and a finite holed plate, is shown to be in very good agreement with available numerical results.