Saline Rinse. A great remedy is to start a saline rinse. There are many out there different types available otc (neilmed sinus rinse). This will help remove the irritatants. Another otc med is that is useful for cough with phlegm is Mucinex (guaifenesin) dm. The Mucinex (guaifenesin) will loosen secretions and the dextromethorphan (dm) will help with the cough.
Treatments... The residual cough is sometimes due to bronchospasm and airway inflammation from the upper airway infection and can require bronchodilator and steroid inhalers to relieve. You would need to see your doctor to determine if this treatment would be helpful for you. The nasal congestion can sometimes be treated with an over-the-counter saline nasal spray.
Try this. I use neti pot for nasal congestion and humidifier treatments with eucalyptus oil. You pour some tepid water in the humidifier and add eucalyptus oil, so you can feel the scent and turn the humidifier on and inhale it in with your nose and exhale it with your nose as well. Works for me like a champ!