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Long hair itself doesn't have a direct relationship with feeling cold during a bath. However, wet hair can contribute to a sensation of feeling colder after a bath, especially in colder environments. When your hair gets wet, it can hold onto water and take longer to dry. If you're in a colder environment or exposed to a draft, the wet hair can make you feel colder as water evaporates from your hair and skin. This is why some people prefer to dry their hair thoroughly after a bath to minimize the cooling effect of wet hair. In general, it's a good idea to keep your body warm after taking a bath, whether you have long hair or not. You can use a towel or a hairdryer to dry your hair and body, and you may also want to wear warm clothing to help maintain your body temperature.

According to your query you want to know about the relationship between cough, cold with long hair.

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No such relation,but one can catch a cold if long hair remains wet for a long time