We consider matching markets at a senior level, where workers are assigned to firms at an unstable matching—the status-quo—which might not be Pareto efficient. It might also be that none of the matchings Pareto superior to the status-quo are Core stable.We propose two weakenings of Core stability: status-quo stability and weakened stability, and the respective mechanisms which lead any status-quo to matchings meeting the stability requirements above mentioned. The first one is inspired by the Top trading cycle and Deferred Acceptance procedures, the other one belongs to the family of Branch and Bound algorithms. The last procedure finds a core stable matching in many-to-one markets whenever it exists, dispensing with the assumption of substitutability.