Look at you cheating with the pneumatic jack!
What you are doing is called underpinning a foundation, so you can Google rabbit hole that til you get the idea you think will work best. I personally would crib the beams next to your column (vertical members), so you have room to pour a new footing under the column. I would also be very tempted to use a screw jack so you can get it perfect, or several different thicknesses of shimming material.
When i bought my house it required us to jack it a bunch, mainly because it originally was built as a mining shack for a coal mine off in the woods close by. They then enclosed a porch to expand it, without fixing the fact that a porch is sloped
Since everything was cobbled together, fixing stuff to a standard that i could tolerate has been rough, and since they were very dumb they also didn't put the foundation deep enough for Illinois, so I'll be doing a ton of this and likely installing a basement one day if i can keep the wife off real estate websites. I already did a mini basement, big enough to hold all the plumbing stuff, maybe 10 x 15. Always fun to go under the house with a skid steer
I need to figure out a way to do stuff like my dad did, he had a pretty good system down. He would somehow manage to ship my mom away for a time, and once it was clear she didn't forget something, the plan would be set in motion. Beers would be opened, buddies would show up, and a few hours later the demolition was so far along you could only move forward
Mom would finally come home, do several "damnit what did you do" and then all would be fine hahahahahahaha. I want a second story here, so if i get the basement in that's exactly how I'll do it.