We study the behavior of the itinerary sequence of each point of the Julia set of \(z\mapsto z^2 + c\) when the parameter \(c\) in the shift locus is allowed to pass through points in the bifurcation locus, which we call narrow. We first show the combinatorial and geometric properties of narrow characteristic arcs. We also show how the itinerary sequence changes in an algorithmic way by using the lamination models proposed by Keller [13]. The algorithm is recently generalized to all bifurcation points by Ishii and Richards [11]. Finally, we found an equivalence relation on the set of \(0\)-\(1\) sequences such that the changing rule is a shift invariant up to the equivalence relation. This generalizes Atela's works ([1], [2]), which dealt with the special case of the generalized rabbit polynomials.