In regards to JOAT, I would say investigate, investigate...go from there. Obviously the flow of the game will dictate the role JOAT plays, but knowing allegiances (good/bad) will be more beneficial to the town than smiting random suspicious players if subtle peeks are left.
That path makes the JOAT's life harder it seems. In a perfect world, I would go:
r1: investigation
r2: investigation
r3: Vig any bad investigation, claim before nightfall, if not investigate
r4-onwards: doc save after claim/vig