Sadie is now almost always given as the full name, so I don't think there's any need to agonize over having a "full" name.
Jack is also very common as a full name and has been for a long time. I know an old man called
Jackie whose full name is
Jacob, so that might work.
Jack as a middle name is all right but you run a fairly high risk of the combo sounding hillbilly. But if you only want
Jack as the middle name, and aren't planning to use
Jack in referring or speaking to him, there's no reason to give a middle name that's any other name than
Jack. If his name is
William Jack, for instance, you call him
William or
Bill or
Will. (Not
Billy Jack; see what I mean about the middle name?) If you want to call him
Jack, just make
Jack or
John or
Jacob the first name all along.