This note advances the work "Simultaneous identification and stabilization of nonlinearly parameterized discrete-time systems by nonlinear least squares algorithm" by Li and Chen by allowing the sensitivity functions of discrete-time nonlinearly parameterized systems with scalar parameters admitting arbitrarily many zeros for large outputs. Stabilizability theorems" and "impossible theorems" have been provided both for time-invariant-parameter and time-varying-parameter cases. A practical estimation algorithm is also proposed to facilitate the implementation in computations.