I have a much better way! Of course, do all of what I suggest with the breaker off for safety.

First, remove the switch from the wall. Take the wires off of the switch and twist them together using a wire nut. This will make the wall outlet always on.

Second, install a ceiling fan remote control and mount it in the wall where the switch used to be. Follow the instructions provided with the remote control.

This is how I've wired my home. The benefit is that the remote control devices will allow speed control, and light dimming. You would not get this by rerunning the wires as you originally asked. Plus you can buy ones that mount in the wall, or on brackets so you can remove the remote and use it from your bedside table.

I don't remember his name, but there's a guy on this board that is a expert at ceiling fan installation.. I'm sure he'd offer good info on how to do the install.