Continuing the same scenario — the screw lodges in the oropharynx, and you've already tried unsuccessfully to retrieve it. What do you do next?

Since there is no description of acute airway obstruction (no choking/inability to breathe — which would warrant calling 911/performing back blows), and your own attempt at retrieval failed, the next appropriate step is still to get the patient to the emergency room for imaging and professional removal, rather than inducing vomiting (which is generally discouraged for swallowed foreign objects, due to risk of aspiration during the vomiting reflex) or simply doing nothing.