Without the Louisiana War (Mr. Hamilton's War for you Demo-Reps) I don't think that the United States would have been able to gain New Orleans.
But maybe even with a dove like Jefferson or Adams in office it might have happened. After all, IMO Hamilton would still be really for it ATL, and regardless of the outcome of the '92 election he was the most powerful figure in the First Republic.
Do you think that the Federal Party would have been able to make itself the champion of the frontier states' without the Louisiana War?