But is Billy Southerner willing to do something that is certain to cost him his farm if caught? Billy Southerner DEFINITELY has something to lose. If he does nothing he will do just fine.
It's a question of risk. In OTL the vast majority of white Southerners weren't willing to risk life and limb to keep fighting the Union (suppressing the blacks is another matter). The risks were to big and the motivation was lacking. Yes they'd prefer independence but that wasn't on the cards and the union wasn't bothering them.
If you have farms being confiscated left and right then Billy Southerner is going to be crapping himself. He might not go out and burn down Tim Yankee's farm but he'll be happy to shelter Johnny Southerner. Now if you restrict such confiscations to a small enough scale that Billy Southerner doesn't start getting worried you are going to have so few farms for Northern veterans that they are going to have to either rapidly assimilate or be driven out by the KKK and White Leagues.
Small Scale Confiscations and settlement by Yankees: They assimilate or get burned out.
Large Scale Confiscations and settlement by Yankees: Terrify white southerners and set of a spiral of atrocities on both sides leading to a bloody quagmire.
Small Scale Confiscations and sale in open auctions: A few carpetbaggers come south and suffer the above but most confiscated land is in the hand of other southerners. The moderates are split off from the die-hards and the South once more becomes a willing part of the Union.