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We consider a nearest-neighbor, one-dimensional random walk $\{X_n\}_{n\geq0}$ in a random i.i.d. environment, in the regime where the walk is transientwith speed v_P > 0 and there exists an $s\in(1,2)$ such that the annealed lawof $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s. Under thequenched law (i.e., conditioned on the environment), we show that no limit lawsare possible. In particular we show that there exist sequences {t_k} and {t_k'}depending on the environment only, such that a quenched central limit theoremholds along the subsequence t_k, but the quenched limiting distribution alongthe subsequence t_k' is a centered reverse exponential distribution. Thiscomplements the results of a recent paper of Peterson and Zeitouni(arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter$s\in(0,1)$.