Augmented Lagrangian algorithms are very popular and successful methods for solving constrained optimization problems. Recently, the global convergence analysis of these methods have been dramatically improved by using the notion of the sequential optimality conditions. Such conditions are optimality conditions independently of the fulfilment of any constraint qualifications and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than the previous stated in the literature. We show that a well-established safeguarded PHR augmented Lagrangian algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a primal-dual augmented Lagrangian variant capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the classical PHR variant when it tends to fail. We show by preliminary numerical tests that, in fact, all the problems already successfully solved by the pure PHR method remain unchanged, while others where it failed are now solved with an acceptable additional computational cost.