If the action isn't so high that the string is being stretched to too sharp, it's most likely the bridge or its saddles are too close to the nut. It may not be an adjustable bridge, in which case there's little you can do. Sounds like it needs moving 2 or 3 mm back, which may well be impossible.
I've had that happen on a lot of guitars but just lived with it since it's not that consequential in that I don't play the low E string that much. But what comes to mind is that the only way you could possibly affect it is to change your relief by adjusting the truss rod. If you adjust the truss rod to get less relief it would help out with the problem.
Necks are wood, and wood changes over time. Humidity can be a factor, temperature can be a factor, etc. The only way you can have a neck that never changes is to have it made out of something besides wood, maybe aluminum or some kind of composite.