This paper discusses the use of a stopping criterion based on the scaling of the Karush-Kuhn-Tucker (KKT) conditions by the norm of the approximate Lagrange multiplier in the ALGENCAN implementation of a safeguarded augmented Lagrangian method. Such stopping criterion is already used in several nonlinear programming solvers, but it has not yet been considered in ALGENCAN due its firm commitment with finding a true KKT point even when the multiplier set is not bounded. In contrast with this view, we present a strong global convergence theory under the quasi-normality constraint qualification, that allow for unbounded multiplier sets, accompanied by an extensive numerical test which shows that the scaled stopping criterion is more efficient in detecting convergence sooner. In particular, by scaling, ALGENCAN is able to recover a solution in some difficult problems where the original implementation fails, while the behavior of the algorithm in the easier instances is maintained. Furthermore, we show that, in some cases, a considerable computational effort is saved, proving the practical usefulness of the proposed strategy.