Question re what to put "under the box"

Poppy is a former shelter cat, with me now for 5 years+. She still doesn't like to be picked up and still "poops" outside her litter box (the latter a longstanding behavior, possibly related to her being declawed by former owners). After several days of trial and error when she first got here, I gave in to her preferred litter box location: on the carpeted, closed in balcony instead of the washable floor of the utility cupboard, where I wanted it.

She poops in the same spot every time, a foot or so away from the box, quite out in the open, comes to let me know she's pooped, and I just pick up and flush her deposits and disinfect the "poop spot.". (We long ago came to terms with her pooping "outside the box." Poppy knows she has "Staff" for a reason...)

She's otherwise become a happy, healthy, bouncy little girl. 

One issue has been with us since Day One: There's a machine-washable mat under the litter box, between box and carpet. (I tried disposables until I hit upon the idea of "machine washable" as cheaper and less "cat-shreddable.") Poppy quite often wets that mat and then goes crazy scratching / scrunching up the mat until I clean the box (almost as soon as she'd out of it) and put a fresh mat down. After much trial and error, I found "Simple Solution" odor spray, and use this on the carpet and wipe the litter box with it, so there's no urine odor detectable. Poppy gets a fresh litter box, with fresh litter, twice a week. 

I never could figure out why / how she wets the mat, until I happened to see her today. She stands in the box to urinate, with her rear end as close to the end of the box as she can get. The litter box is 24 x 16 with 6-inch walls. A longer box wouldn't help ... I suspect she somehow feels more secure backed up against the end of the box. Would a box with higher walls stop the peeing over the wall behavior? 

Money is tight, so I don't want to buy something in the hope that it works. I can go on doing a Poppy-load of laundry every week , no problem, but wondered if there is a simple solution somebody else has come up with? 

Storage tubs. Many of us use them. Lots of sizes to pick from. Personally, I cut an entrance hole in one side (about 5 inches from the bottom, up to the top rim), but some people don't and their cats are willing to jump in. I use the 18-gallon Sterilite tubs, they're about $5. But look around at all the sizes and shapes and see which one you think will work best.

i put an extra large dog crate tray under HD's extra large cat box.  we put a layer of litter on the crate tray so if HD feels like going outside the box it's not a problem, and under the crate tray is a small carpet that extends so HK wipes his feet as he leaves.

Thanks to both for replying. The storage tubs (higher walls than an ordinary litter box?) are the same price here (Canada) as a high-walled (9 inch) litter box, so I guess price-wise it doesn't matter if I get either.

The second option ... the dog crate tray ... seems pricey at first, but looks easily cleaned, so maybe I'll try that under the litter box. Gives me some ideas anyway.

Thank you! 

we paid around $40 for the tray, but we've been using it for several years, and when HK started having litter box issues, he usually went on the tray instead of somewhere less fortunate, so i feel it's paid for itself.  in fact, this morning he had some discomfort and went on the tray instead of on the couch or my bed.

I have had large cat missing the edge problems before. if the litter box is inside a high walled tote it will catch over spill. But also she may feel safe, and not move to the edge so much. Have you tried a really low sided box with washable mats in it?