There is usually a lacquer on the top to protect the dial, painted text and so on which is probably what is lifting. It might be possible to remove that and replace with new but it can lead to loss of all the text and no-one can say for sure if it will be possible - sometimes it works and other times it doesn't. I don't like doing this and most watchmakers will be the same and insist that you accept you may lose the dial.
Where are you and do you have a watchmaker that you use? You need someone who has done this before and not everyone will have - I'm almost retired now and not taking on work but there are other vintage specialist watchmakers who should be able to help.
Al,
@Archer may be able to say if a maroon (burgundy) dial for a 566.0024 stainless steel case is available in the Omega network. I suspect not...
Cheers, Chris
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