You can't catch a cold by going out with wet hair. A common cold is caused by a virus - of which there are many - and you can only catch one of them from another person who is infected. You can breathe in the virus if someone has sneezed into the air you breathe or you can contract it by touching a surface (worktop, eating utensil) which has been contaminated by someone with the virus, and then touching your own nose or mouth.
I believe that the "you'll catch a cold if you go out with wet hair" warning has evolved from the earlier "you'll catch your death of cold". In the days before central heating and warm water on tap, if a person got wet and went out into cold temperatures their body temperature could drop dramatically and it would then be very hard to warm up again. In the "olden days" many people would literally "catch their death of cold" and die of hypothermia. It was the cold temperature that killed them, not the common cold virus. Somewhere along the line catching an actual viral cold and becoming hypothermic and pegging out have blended together.