Let $χ^λ_μ$ be the value of the irreducible character $χ^λ$ of the Hecke algebra of the symmetric group on the conjugacy class of type $μ$. The usual Murnaghan-Nakayama rule provides an iterative algorithm based on reduction of the lower partition $μ$. In this paper, we establish a dual Murnaghan-Nakayama rule for Hecke algebras of type $A$ using vertex operators by applying reduction to the upper partition $λ$. We formulate an explicit recursion of the dual Murnaghan-Nakayama rule by employing the combinatorial model of ``brick tabloids", which refines a previous result by two of us (J. Algebra 598 (2022), 24--47).