Put a few disposable hand /wet/alcohol based wipes/sachets (i.e. the sealed ones you get from fast food restaurants that have a chemical smell) in your saddle bag, to use clean your hands if you have to put your chain back on the gears, or after any other dirty work you might need to do on a ride.

Unfortunately I worked this out the hard way yesterday, and my white handle bar tape now isn't as white as it used to be.