We study the tunneling tail of eigenfunctions of the quantum map using arbitrary precision arithmetic and find that nonmonotonic decaying tails accompanied by step structures appear even when the corresponding classical system is extremely close to the integrable limit. Using the integrable basis constructed with the Baker-Campbell-Hausdorff (BCH) formula, we clarify that the observed structure emerges due to the coupling with excited states via the quantum resonance mechanism. Further calculations reveal that the step structure gives stretched exponential decay as a function of the inverse Planck constant, which is not expected to appear in normal tunneling processes.