A warm plunge might help more. A cold punge will cause your body to heat up again instantly to adjust be body temperature.

Cold plunge re-alkalines the blood and helps reset your system but not good if sick. Body is warm and using inflammation to fight off whatever has ya under the weather

Body heat kills some viruses.

Depends on the illness but generally your body heats up to make things hard for the virus/bacteria.

A cold shower may be a better, less intense option to start with