Different brands and styles of snuff will have differing levels of messiness, complicated by how much is being used (as an ex-smoker, I use a hell of a lot in comparison to some others), but we all have encrusted nasal passages and moustaches if we don’t keep it cleaned up with a handkerchief or Kleenex.  I can run through two handkerchiefs in a day.

Over time, the tobacco will naturally slide down and out of the nostril, requiring wiping away.  We also (usually) blow the older tobacco out of our noses before adding more.  It is a balancing act, really.  If you keep at it you will find a natural rhythm that works for you.  Hopefully, this rambling helped a little