Harman, you have many alternatives...
...but you might find the approach I used to be inexpensive, easy, presentable and efficient. Consider using isocyanurate exterior foam board, usually foil backed on both sides and sold at Home Depot and Scotties in 1" thicknesses. It's very inexpensive, can be cut with a utility knife and can be inserted in a staggered fashion within the old box (leaving the existing liner in place) using Liquid Nails or some such adhesive. Once you've built up the box (either after tearing out the old stuff or just adding to it), use either PVC sheet or ABS sheet plastic for your new icebox liner. It also can be cemented in place. I used 3/16" PVC white sheet which is shiny, easy to keep clean, easy to cut & fit, and the seams can be bedded e.g. with polysulfide. To seal all the seams, I purchased some el cheapo PVC trim pieces intended for use with interior wall linings, also found at Home Depot, and which is applied with PVC cement. The joints are waterproof and the end result looks very nicely finished.Jack