A couple of remedies you can try:
Chew 4-5 tulsi leaves the first thing in the morning. You can continue taking Tulsi leaves to boost your immunity.
Take a teaspoonful of honey before going to bed at night to reduce the severity of cough.
You can also try Sandu Mahasudarshan Kadha an effective immunity booster. It helps manage flu like symptoms like headache, cold, cough and regulates body temperature.