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I would like to in-lay a zoom-ed crop of my photo within the main photo. The in-layed photo would be on a free part of the background, just to show-case the detail.

Is the rectangle tool + layer with copy the best option for this? What is the best way to do this that preserves the most detail?

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One way I do it is the opposite approach - I scale down the main photo to the final size it will be used and keep the cropped version at full size (for example on a separate temporary new file) and put it back into the file.

I also like to turn these layers to linked "smart objects" so I could adjust sizes without losing pixel information. Smart objects are best used for shrinking down, because you can't control pixel interpolation when scaling up.

Also this method is most suited when your photo resolution is larger than your final file or print requirements are. You can increase the size too, but keep in mind the quality. Try it out and find what suits you best!

Illustration of cropped out "zoomed in" part of photo

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With creating the layer for the zoom-in, selection and layer via copy will probably be fastest as you have found.

As for preserving detail with scaling, that will mostly be in the method in which you scale it. Because the image doesn't have enough pixels to freely scale it up to the size you want, your program will need to "guess" at what they pixels would be, taking an average of what color the surrounding pixels are. This process is called interpolation, or resampling.

In Photoshop, Bicubic Smoother looks best for enlargements. You can either change to this while in free transform mode, or change this to the default scaling in your preferences. Bicubic automatic should work best is most cases though.

In Gimp, your options are more limited, but cubic or Lanczos will look slightly better, because they take into account more of the surrounding pixel data than the lesser interpolation methods.

Take a moment to try each of the scaling methods, or look up some website that has. Photos tend to use the higher levels, while pixel art will almost always use nearest neighbor (no interpolation).

enter image description here

If you want more clarity than that, you'll need to use some other method to sharpen the image, or find a higher quality image. Photoshop can work magic, but as the saying goes: garbage in, garbage out. I would suggest looking at the photoshop filters, or look up the high pass overlay method to enhance edges.

Read more about interpolation here under resampling:
https://helpx.adobe.com/photoshop/using/image-size-resolution.html

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