If you are putting in tweeters I think it's more complicated than just running a wire.
Right now all the sounds are going to the door speakers. Bass and tweeters...not a good thing.
Plus, if you don't swap out the door speakers and add a quality amplifier, I really don't know if the sound will be a lot better. I swapped from front door speakers and tweeters for focal speakers and a focal amp...and turned off the back speakers. The sound in my car is really awesome to my ears. Well worth it!