You could easily lay deeper shelves on top of existing shelves for easy fix. Depending on the widths of these shelves, you may even find ready made melamine shelves in 12" depth and 3' or 4' lengths and work with those . Easy fix.
You can work with a closet company or even a Shelf Genie, to see what options you would have if the budget is there for it. It would be REALLY nice if you could remove the curtain and have "cabinet fronts" on pull out drawers. Fridge may prevent that?
I would start by purging the closet and then see what is really needed to keep. Start an idea book here on Houzz for small pantry closets. You are so sweet to help.

What is the dimension of the entire space? What is the budget? What is the expected time frame, a couple years or the foreseeable future, hopefully decades to come? Would an investment in organization also add to the expected market value of the house?

I like using 1x wood or high grade plywood with covered edges for shelving rather than buying commercial organizers unless you get lucky and find some which fit the space with only tiny gaps. I would look at a mixture of fixed and adjustable shelves but honestly, most people dont actually adjust shelf spacing after the initial install. In 26 years, I have changed shelf spacing maybe twice, by adding additional smaller partitions.

My irritation with most pantries is the shelves are too far apart. I added a shallow shelf at eye level which is only a few inches high, for small cans and pudding boxes. If I have the occasional tall item it goes in front

My initial thought is deeper shelves across the back, maybe 10 to 12 deep with plenty of height at the top for a multipack of papertowels and the lowest one with room under it for two liter bottles. On the left wall, a series of shallow shelves of differing heights for single layer of cans, spices, perhaps a couple of small baskets for packets. I would see that as the depth of the door framing. On the right side, making an ell with the back wall, I would use baskets hanging on the wall for all those little things that clutter up spaces like plastic container lids or picnic ware.

How talented and/or adventurous are you? Moving the doorway to the right hand side of the pantry would allow you to rehang a solid door with storage on the back. That would be a big increase in functionality but with a major increase in effort required. You might also get similar results with an investment in a counter depth fridge

Nice you are helping another.

An easy, cheap soluion is to at least deepen the back shelving to at least 12" deep. 15- 18" ok too.

Please draw a sketch of the inside, with measurements so we can see how much space you really have.

What are your MIL's goals? Budget range?

I would start by taking everything out. Remove any nonfood item that has not been used in the last six months, such as the blue plastic guitar. Remove any canned things that are not getting used up and replaced regularly. Then you'll have a clean start with only useful things to store.

The way these are constructed will make it difficult to rip them out and start fresh. Not impossible, of course, but who knows what you will find behind the side panels? Does the small furniture piece need to stay?

Without dimensions I'm only guessing, but it seems the most efficient use of the space would be shelves just across the back. Make them 16 - 18" deep, for large, bulky items, spaced to hold things that will actually be stored there (the tub on the floor, perhaps paper towels on the first shelf)

Next fill the remaining space with 12" shelves - twice as many as you have now, with various spacing. Some will be as tall as canisters and cereal boxes; others will be as tall as canned goods; others as tall as bottles. When things are grouped by height, you have no wasted space between the top of an item and the shelf above - and you avoid stacking things, making it easier to find and use things. Bins or boxes can be used to keep small, like items together.

Any remaining space between shelves and door frame on the right can be used for hanging a broom and mop.

Good luck with your project!