Drive a skid-steer loader or excavator up next to the chimney wall. Use the machine to insert a helical pier into the ground. This is a long screw that goes down into the soil until it hits a surface that will give it the stability to hold up the chimney. Install as many as you need to raise the chimney, but at least two are recommended.
Slide at least two hydraulic jacks into the space under the bottom of the chimney. Raise the chimney to its level position using the jacks.
Apply exterior caulk to seal the chimney back into place. Adjust the helical piers so they support the chimney in this position and remove the jacks.