With boots 6" or less I always get dirt down in my boots with out gaiters. The gaiters also help keep socks clean. So that is why I wear gaiters, short Outdoor Research Gaiters.
Pick up your feet when you walk and carry an extra pair of socks!
I'm not seeing the value of something else to put on.
As freestone points out, feet get dirty regardless, and there are always points/stops where you can take off your boots and let the feet breathe. That Noontime dip in a stream is really good for the feet and lets the boots/socks air out.