I found a solution to this here. Using the OP’s example you basically run
stdbuf -oL /homedir/MyScript &> some_log.log
and then the buffer gets flushed after each line of output. I often combine this with nohup
to run long jobs on a remote machine.
stdbuf -oL nohup /homedir/MyScript &> some_log.log
This way your process doesn’t get cancelled when you log out.