Shaving soap may save your life.
Nobody is quite sure what keeps a reanimate moving, and why they aren't just rotting piles of meat. They're perfectly intact, and will even heal wounds over time. Nothing like those half gone grey skinned creatures from old movies. More impressively, this oddly good health is in spite of having next to no sense of hygiene. Far as we can tell, they don't bathe, and change clothes only when what they're wearing has ripper or rotted completely off, if that much. Death be not proud and all that. (Though I don't think that is what the poet meant...) If you have a good nose and aren't too ripe yourself, you can smell them on the wind sometimes.
Dogs, of course, are a more reliable warning system though you have to be careful. To start, barking and whining can reveal your position. Furthermore, you will probably get a lot of false alarms. What few tests have been conducted seem to show that an animals aversion to reanimates is just based on them seeming like humans'. A wild horse can't tell the difference between a not quite alive person and a not yet dead one - it just doesn't like people. So the dog that barks at reanimates, is really just barking at any person that isn't you.
It might seem like a vanity, but keeping your hair nicely styled and getting rid of that matted beard is a good way to avoid getting shot due to mis-identification. If you're shot because they can tell who you are, well that is your own damn fault, not mine.