Sounds like a good plan so far. A couple of additional possibilities - have two letters from the new place of employment - one very formal one from the human resources department spelling out all of the required data, and a second one, from your boss or department head (also on company letterhead but personally signed) that is more of a personal "vouch for" type of reference, maybe even stating your importance to the company (to remove any doubt that you're going to overstay your visa to work in Italy).

Overdress a little - nothing too extreme.

Keep the explanation of what went wrong with the first attempt to a very concise minimum. Choose wording that "looks forward", not back - especially avoid any language that in any way smacks of recrimination.