Don't know if you can move the plumbing but maybe an expert could "pipe" in (see what I did there? Pipe?)

Anyway...I would say that you could add shelves to the back since the drawers don't go all the way in...You would have to measure your space to find the correct cans...do you need recycling and garbage..

Good question...I'm always trying to make the most out of the under sink space...

You could replace the sink with one with a drain at the rear/side.

Would making a "U" in the cabinet to go around the pipe work for what you are trying to achieve? Something like this?

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I didn't watch this but it may help in giving you some direction. You just need to find someone with the proper equipment that also includes the ability to cut the metal.

https://www.youtube.com/watch?v=q7S9e8z5zNw

The small amount of extra storage space I'd get there would not be worth - to me - the cost and trouble of moving the piping or changing the sink. I'd get one pull out drawer (which I think you have) and add different sized storage boxes (some shallow, some deep) to help keep things organized and separate. (Even something like these from Ikea, so you won't care too much if they get stained or damaged.) Line the drawer with a textured shelf liner to keep the boxes from sliding (or use a little velcro to keep them in place). You could add some hooks to the sides of the under-sink cabinet to hang things too, like a hair dryer or small storage basket. Easy, fast, inexpensive, no risk of a lousy plumber causing damage.