Since the Army of dead breached the wall and is marching south, I've been thinking. Could White Walkers reanimate corpses/skeletons buried in cemeteries? This would increase numbers of wights immensely, especially when they enter southern parts of the realm.
It would appear so (at least in the show). The wights that attacked and killed Jojen Reed appeared to come from graves:
It is not yet confirmed that Jojen is dead in the books, but it is heavily implied.
It is also highly speculated that we will see Stark dead rising from the Crypts of Winterfell.
From our first contact with the Wildlings, it is stressed that dead bodies should be burned, not buried.
It would likely depend on the state of decomposition of the bodies. While we have seen some very skeletal looking wights, they still have some form of tissue holding them together.
If burning a body works to save you from being reanimated I think a body that is just bones will remain that way. A standard fire isn't going to destroy bones.
Update: I was wrong.
In S08E03 we have 100% confirmation.
in 0:45 we can see the NK resurrecting corpses around, in 3:00 we can see wights in the crypt.
OP: "especially when they enter southern parts of the realm"
well, thats not gonna happen...