The best way to get rid of a chesty cough will vary depending on the underlying cause. However, some general tips to help relieve a chesty cough include drinking plenty of fluids, avoiding irritants such as smoke and dust, and using a humidifier. If the cough is due to an infection, then treatment with antibiotics may be necessary. In more severe cases, such as a chronic cough, medication may be needed to provide relief. If the chesty cough is becoming bothersome, it's important to discuss it with a healthcare professional.