There is no reference to any need for it to be a "woods tile" clearing.
Here's what a search of the 3.1 rules PDF returns when you search for "Walking the Woods":
7.3.4 Walking the Woods: When transformed into a beast that
“walks the woods”, a character or denizen can move from
clearing to clearing within a tile without following the roadways.
He can record and do a Move to any clearing in his tile. He must
follow the roadways normally when he moves from one tile to
a. When he moves, his horses move with him.
b. When a beast moves from clearing to clearing, a follower
can follow it only if he is also a beast or if the clearings are
connected by a roadway. If the clearings are joined by a hidden
path or secret passage, everyone who is following the beast