I learned the secret from my father-in-law because we were forever having to pull driven posts used to mount a dock. I have used it for livestock fencing and it works every time. It is not too different from your idea of a farm jack and chain.
Use a come-along, a heavy 2" steel pipe and a short lenght of heavy chain. Wrap the chain around the T-post. Hook the come-along on the top of the pipe and support the bottom of the pipe on a board to keep it from being pulled into the ground. Hook the other end of the come-along on the chain wrapped around the T-Post to be extracted. Ratchet the come-along and pull the post out of the ground and go on to the next one.
Once you have the system down you will spend about 10 minutes per post.