Foundation raising on old walls or new ones
Some raises extend the existing foundation, and others need a new one built underneath. A basement's underground walls usually have to be rebuilt or filled. If the design professional finds the old foundation sound, extending it can keep the job smaller. If it's cracked or questionable, I'd rather put the money into new foundation work than into anything you'll see on top, because a nice finish can't make up for what's underneath it.
What shapes the cost of foundation raising
What the house sits on drives the price most. A home on piers and beams or over a crawl space is usually the easiest to raise because there's less digging, a slab is harder and a basement costs the most. Lift height and repairs found along the way move the price too. A lift-only price leaves the utility work and new foundation to separate contracts, while a turnkey price folds them into one number, so ask what each bid covers before you compare totals.