It's really not all that difficult.
You could hold back from the original foundation on 3 sides about 3'-4' & install 4x12 Beams we spanned I believe 22' long by 16' deep with 4 or 5 4x12 Beams & steel posts.
We started by getting one inside of the end foundation that was going to be knocked out this supported the house while we removed that portion ot the foundation. This opened up the whole end of the under side of the house. Then we dug back in a ways from the short posts & replaced them with full length dirt floor to ceiling posts. Then we dug back a ways & installed another 4x12 & posts & repeated until we were as far back under the house as we wanted to be. Almost all the digging was done by a tracked end loader. We had already dug out for a big living room that was to be added to the end of the house. So we had plenty of room to maneuver the end loader. We even used the end loader to hold the beam against the bottom of the existing stringers. as we set the jacks in place to raise it to its final position & install the posts.
The beams we installed ran across the opening & the pre-existing stringers ran the other way.
By holding back from the original foundation on 3 sides it makes it simpler than having to raise the whole house. The house stays in place & you pour concrete wall about 3'-4' high & stud up to the bottom of the existing house. You could also pour a footer & lay up block if your a glutton for punishment.
"Forget the flat stuff slap something on the spinny thing and lets go, we're burning daylight" Bart Leetch
"If it ain't round you may be a knuckle dragger""Turners drag their nuckles too, they just do it at a higher RPM"Bart