Implemented my solution for a phone holder today. I used the OP's idea of reusing parts of the cupholder. I purchased a spare cupholder off of ebay and removed the top hoop as shown in the original post.
Next I purchased a mounting disk off of Amazon that attaches with 2 sided 3M tape.
Mounting disk
I attach the disk to the bottom via the 3M tape.
I also had an iOttie phone holder that I've had a couple of years and attached it to the disk.
iOttie Phone Mount
Here is the final result: