This likely has nothing to do with internal components, and there's no need to send it to Omega - in fact if the OP wants it to remain looking vintage, that's the last place he should send it.
When a hand is not resetting to zero there are a few common reasons, and I'm listing them from most common to least:
1 - It wasn't set in the right position when it was mounted.
2 - The hand's tube has slipped on the post.
3 - The rivet that holds the flat part of the hand to the tube has failed.
While you might expect that if the hand's tube has slipped on the post it will continue to slip, that's not always true. So it is possible that it has slipped on a big reset (when the hand is moving from it's farthest point to the reset position where it has the most inertia) and now if you are only doing smaller resets where the hand doesn't travel far, it would stay in the new location. If it continues to slip it will slip in the direction of the reset.
If the rivet is loose, then the hand will reset to very random positions.
The only way to know for sure is to get the watchmaker to look it over. If the tube on the hand is loose, it can be tightened. if it has split, then it's a bigger repair and you could get a new tube fitted to the hand if keeping the hand is important.
Cheers, Al
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