Their draft position isn't going to move that much, though. In the AL, they're 3.5 behind Detroit and 5.0 behind Cleveland. They have a chance to catch either or both but mostly only because they play head-to-head again. But the Twins are 6.5 behind the Angels so that's not happening.
In the NL, they might catch Colorado, as it appears they're two games back, but after that it's the Mets and they're 6.5 back of them, too.
Even if the Twins play well and go over .500 down the stretch, it's unlikely they pass more than two, maybe three teams. In the end, we're only talking about the difference between picking around eighth or around eleventh.