Photoshop :: How To Professionally Delete Background Of An Image

I am working on this book cover in Illustrator, and I am having problems deleting the background of this one image (in ps). I use the magic wand, but it doesn't look smooth me.

I have this image of some rings on a white background I got from stockx or one of those sites. These rings need to sit on an image (which is a large monogram M). I am trying to make the background transparent, and my attempts....well, they are ok for the average eye, but for something going to print, it just is not good enough. I am still learning illustrator, so I am trying to delete this background using photoshop. I used the magic wand to delete the majority, and then tried to use the eraser to clean up. Still looks like garbage to me.

I became so frustrated I thought about about a diff approach from deleting the background. I would just create the "M" in one layer, dupl the layer, crop it, etc etc. Only prob is when I try, transferring back to Illustrator makes the image jagged. And trying to crop in Illustrator? Well that is like calculus to me.