What's the best choice of lens from an optics surplus shop to get a Stanhope Lens-like magnifying function?

Most magnifying lenses will reduce the focus distance therefore increasing the magnification. You really need a lens designer, which you might find on physics SE, but since they won't know what is inside the phone, they may not be able to help. Your best bet is trial and error.

iPhone8 Only:

iPhone8 with expensive 8X Loupe

iPhone8 with cheap magnifier. The 45X (smaller lens) in https://www.amazon.com/gp/product/B075WRJPFN Something like this is probably close to what you want, but it is not a Stanhope, it is not in contact with the object. It is recessed into this magnifier so this is the closest that I can get the object.

I don't know what your overall goal is, but consider buying a macro lens attachment for your phone. Sorry, I don't have one or I would test it for you.

Because your plan is to mount a supplemental close-up lens, in direct contact with the surface of your camera phone, you believe this added lens must have a very short back focus. This is incorrect thinking. In fact, you can procure most any hand-held magnifier and place it in direct contact with the phone. The air-space between the phone and lens will not have that much influence as to the outcome. A magnifying lens placed before the camera lens, acts almost exactly as reading glasses perform. Best would be a jeweler’s loupe. I suggest you procure a Hasting’s 10X loop. The Hasting’s is a three element color corrected magnifier favored by jewelers, stamp, and coin collectors. This would be your best bet.

The 10x or 10 power translates to the distance eye-to-object or in this case, camera-to-object. A 10X magnifier shortens this distance by a factor of 10. The normal comfortable close viewing distance is about 10 inches. Mount a 10x and this distance shortens to 1 inch; hence the object is seen as 10x larger as compared to the unaided eye (camera).