in the basement either by pouring a footing, building a strong decking from rail road ties or some other foundation that will not sink and will hold the weight.
Then once this is done, use your floor jack and a 6X6 Post pushing on the compromised joist. You may want to push on the two or three on either side of the questionable joists to spread the pressure around some.
I would definately not screw or jack up the joists the full amount in one setting. I would go 1/2 inch per day until I am there and test each increment the next day to see if your footing is holding. Going all the distance at once could cause some stressing and more problems.
Also, for future adjustments. Use a screw jack and keep enough turns available for adjustments. fly your flag.