Raising the Wall
Unscrew the nut from all anchor bolts connecting the sill plate to the concrete with a wrench.
Connect the studs to the top plate of the wall with galvanized-steel fastening plates. Set a metal plate on the surface of the stud, overlapping the top plate, and nail it with 1-inch nails. This prevents the top plate from coming off the studs when you raise the wall.
Set a hydraulic jack on a piece of scrap wood next to a corner stud and set a lally column on top of it. Adjust the column so it fits tightly against the top plate next to the stud, then pump the jack to raise the wall a fraction of an inch. Be careful as you do this; there will be many creaks and groans, but listen for any especially loud ones that might signify something breaking. Continue pumping until the sill plate is off the floor. You will know when this happens because the wall will move when you push on it.
Place a lally column on the other side of the stud. Rest the column on a steel plate on the concrete and place another steel plate between the top of the column and the top plate of the wall. Adjust the nut of the column with a wrench until it is tightly situated and holding the wall securely. Then lower the jack and remove the first column.
Place three or four more columns in the same way along the length of the wall until the entire wall is raised.
Removing the Old Plate
Cut the nails holding the studs to the sill plate with a reciprocating saw fitted with a steel-cutting blade. Do this by sliding the blade between the bottom of the stud and the plate and sawing through the nails.
Go outside the garage and remove enough siding to expose the entire sill plate. Then free the plate from the concrete by sliding the saw blade between the plate and the concrete and sawing through the anchor bolts.
Remove the old plate by either pounding it from inside with a hammer or prying it from outside with a pry bar.
Installing the New Plate
Slide a pressure-treated board into the space left by removal of the old plate. Use lumber of the same dimensions as the old plate.
Attach the studs to the new sill plate with galvanized-steel corner brackets. Nail the brackets to the wood with 1-inch nails.
Lower the wall by placing the jack next to each lally column in turn, raising the wall slightly, removing the column, and lowering and removing the jack.
Drill holes through the new sill plate and into the concrete with a hammer drill fitted with a 3/8-inch masonry bit. Insert 3/8-inch lead lag shields into each hole, then screw in 3/8-inch lag screws with a wrench to hold the new sill plate to the concrete.
Replace the siding you removed on the outside wall of the garage.