In this paper, we establish a different approach for solving the system of three coupled two-sided Sylvester-type quaternion matrix equations $A_i{X}_i{B}_i+C_i{X}_{i+1}{D}_i=E_i,i=\overline{1,3}$. We give some new necessary and sufficient conditions for the existence of a solution to this system in terms of Moore-Penrose inverses of the matrices involved. We show that these new solvability conditions are equivalent with the solvability conditions which were presented in a recent paper [Linear Algebra Appl. 2016;496:549–593]. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general η-Hermitian solution to a system of quaternion matrix equations.