I've been regularly using stacking for a few years now, and also often encounter this problem. As you said, the issue is the large distance between the focal plains. When the background is in focus, the acorn is blurred and expands further than when it is in focus, covering part of the background. So there's a 'dead' area with no relevant information for either. It becomes increasingly worse with higher magnification and wider aperture.
I use Helicon for stacking, and in cases like this I use the Retouching tool to manually brush in these areas. This does not completely solve the problem, but improves the result. If the 'halo' is still very noticable I use the spot removal tool in Lightroom to replace the blurred parts with a suitable part of the image, usually close by. If this isn't possible I just try to adjust the clarity/contrast in that area to make it less noticable and better match the rest of the image.
I've attached two 150% crops of an image to show the difference before and after the spot removal in Lightroom. Since I just did it quickly right now it's a little crude. There may be better ways to do this in PS, I'm far from an expert in post processing/retouching as I try to do the bare minimum necessary. If other users here have better ways to deal with it I'd certainly be interested as well.