Pumpkin seeds (the hulless type are best, but can be trickier to grow) run about 40% oil. I always wanted to get one of those neat little oilseed presses, but could never justify spending the money. I suspect if I really wanted/needed to press vegetable oil, my son's could cobble something up using a hydraulic press with some sort of a heat source that would work, and in large enough quantities to make sense (the biggest drawback I can see with the tabletop oilseed presses)
But we'd be going back to lard soap... wait, I never stopped! I've made many, many pounds of it over the decades, and the longer it ages, the better it becomes... any objectionable odor (which shouldn't be a problem in the first place as l9ng as you use fresh lard) vanishes, and it becomes milder with age. I usually let it air out and dry for 3-6 months, then seal it up for storage, as leaving it unwrapped over long periods of time dries it eventually into a rock hard lump.
I recently found a 2 1/2 gallon plastic ice cream tub (Schwanns company used to sell ice cream in 6 quart and 10 quart tubs which were really heavy, durable plastic. We haven't bought anything from them in at least 25 years, and we still have half a dozen of them around!) tucked away in a corner of the basement, under a work bench. It was full... I just didn't know of what! I cautiously opened it, and discovered it was full of homemade soft soap... had to be 20 years old! It's currently in the barn milkhouse, where we're using it for cleaning various items too unwieldy to wash in the kitchen sink. It smells wonderful (it was unscented) and works amazingly well. Oh... when we had our house fire back in 1980, we tried *everything* to remove the smoke stains and odor from everything from clothing to hardwoid furniture. The ONLY thing that worked was homemade soft soap!
It's easy to add fragrance in a very low tech way by simply sealing the bars with fragrant plant materials... lilac blossoms work really well, as does lemon balm leaves, peppermint leaves, or pretty much anything you can find which appeals to you.
The scent isn't as long lasting as using essential oils, but it does work, and you'd only use that technique for special "toiket" soaps. Most essential oils, even if you had stocked some, or has enough plant material to distill some, would be far too valuable as medicine to use for aesthetics. Although I always try to keep quite a bit of tea tree oil around, as adding 1 ounce of it to 1# of soap is a very useful preventative for jock itch, ringworm, female fungal infections (external) and athletes foot. We keep a bar in the shower at all times.
Hogs, especially the older breeds (which were valued as "lard hogs" for precisely that reason) can produce a LOT of lard. My son butchered 7 pastured young hogs last fall (supplemented with corn, pumpkins and garden waste) and ended upnwith over 300# of lard! Lard is a valuable (and tasty, if properly rendered and stored) cooking fat, but few families are going to need more than 100# in a year, especially if they have a dairy cow or two for butter, and butcher a few dozen old laying hens for soup and stew... you get a LOT of chicken fat off a few old hens. Schmaltz (the Jewish term for chicken fat) may be the tastiest fat ever for making dumplings, biscuits and other savory pastries. It's also excellent for frying, sauteeing vegetables, and almost any non-sweet type baking use. It does tend to bee too soft to make good flaky pie crust or layered pastries, but lard or butter- or a combination of the two- are the best for those anyway.
Anyone with the ability to raise livestock shouldn't lack for fat in their diet or for soap. If you're going to have to rely on your gardens for most of your food, it will be trickier. Still, raising pumpkins (probably in a waste area, to avoid using too much garden space), using the flesh for food (or animal feed... they also make excellent deer food, and would be a great way to lure deer or wild hogs within arrow range of a blind or stand) and the seeds to press for oil might work.
Setting snares for raccoons and woodchucks to protect your garden crops, or simply lying in wait with a .22 or good pellet gun would give you some fat (both those tend to be quite fatty, especially in fall) , and it would probably be better in soap than soup... "wild" fat often can be pretty gamey.
I'm sorry I don't have an answer to the OP... I've never used milk to make soap.
Summerthyme