"They agreed by November 2005. But the accident that/the same month caused tension."
Should it be "that same month" or "the same month"?
Either "that" or "the" may be used. "That same" draws attention to the surprising or significant fact that it was the same month that the thing happened (e.g. they agreed surprisingly or notably quickly), while "the same" is neutral and makes the statement merely informative.
The police relied on statements by a neighbour but it later turned out that the murder was committed by that same person.
I resigned last Friday and left that same day.
Other ways of saying "that same" include "the selfsame" and "the very same".