Looking at MLBTR's article on the White Sox. I looked up the outfielders.
Dexter Fowler looks like a solid second choice if you aren't going for a power bat.
.750-.770 OPS his entire career including a .774 OPS in Houston. So there is at least a baseline for if he will be able to do it in the AL.
OBP, .363 in his MLB career. I usually don't like this stat for power hitters. But...
Almost all his starts have been #1. So he's a pure leadoff hitter.
He steals some bases, but not much. Then again, that's the angle the Angels have gone to anyways.
He'll be 30. So probably a 4-5 year contract? And as the article pointed out, $8-15 million a year.
This guy IMO for price, length, need, and risk is a much better option then Heyward.