Hi R.,
I use Dr. Bronner's Castile soap to wash my hair. I was having a hard time with ordinary shampoos leaving buildup, or stripping too much so my natural oils would overproduce. Since I've started using Bronner's there's a big difference. I rinse everyday, but actually wash about once a week. You could wash more often, but everyday leads to over-drying. The day I wash, my hair looks like its straight out of a Pantene commercial, lol, and is a bit more 'fly-away' than I'm used to. The next couple of days its happy and healthy. When it gets too oily, I wash it.
Bronner's is 100% vegetarian - all vegetable oil based, not animal tested, completely biodegradable, and you can not only use it on your hair, but just about everything! It comes in Baby-mild Unscented, which is truly unscented - amazing!, and I think 5 scents, all of which are essential oils - no artificial "Fragrance".
In response to mouthwash: I suppose you could go that route - the alcohol in the mouthwash would certainly remove buildup from your hair and stimulate your scalp to produce more oil, but I guess I wouldn't want to walk around smelling like Scope all day, lol.
Good luck,
Ray