A 62-year-old woman with a history of epiglottitis (now resolving on antibiotics) and significant facial edema presents for urgent airway management. She is conscious, alert, and maintaining SpO₂ 94% on room air. Stridor is absent at rest. The anesthesiologist decides to perform awake fiberoptic intubation. During the procedure, after successful passage of the fiberscope into the trachea and visualization of the carina, the endotracheal tube becomes stuck at the level of the vocal cords and cannot be advanced further. What is the MOST appropriate immediate action?