What you need to show is not that H is necessary but that E is, and you haven't done that. In order to deduce that E is necessary, it is necessary to show that the Ci are all necessary. Failing that, you have an obvious modal fallacy. If T is the time at which E occurs, it may be true that no causal agent could go back and undo the Ci, bring it about they didn't occur, or prevent them from occurring. It doesn't follow that the Ci are necessary.
E is necessary only if it is necessary at the time it occurs.
Consider, again, the event of Jack's grinning at T1. That is necessary only if it was physically necessary at T1 that Jack grin. Hence, Jack's grinning at T1 was necessary. And in order to infer that physical necessity attaches to all human actions you need to suppose that physical necessity attaches to all the antecedent events. I'm suggesting that the argument offered for this assumption is fallacious.