If an Outpost is played during a Possessed hand, is the Outpost hand played next?
The turn Outpost is referring to is "this one", the Possession turn.  Outpost doesn't really care what turn it was played on.

So during the clean-up phase of the Possession turn, the possessed player will draw 3 cards instead of 5, and immediately take their Outpost turn.

At the end of the Outpost turn, the originally possessed player will draw 5 cards as normal.  Turn order then returns to where we were before the Possession was played, so it's now the possessed player's turn (finally!) and they take a normal turn.

So your opinion is correct.