I could see that. Or the new Nashville team affiliates with a new New Orleans AAA team. Indeed, I would think the relatively short travel distance between Nashville and New Orleans aligns well with the stated intention of reducing travel distances between MLB teams and their high affiliates. Either way, if they can get a new (or heavily renovated) park, New Orleans is a prime choice for a AAA team. If that were to happen and Nashville were to affiliate with New Orleans, Milwaukee could conceivable grab Wichita moving up to AAA, being that Wichita was supposed to be a AAA team only for them to get squeezed to AA before they got to play a AAA game. But of course that would mean at least 2 new AA teams would need to be created. One to affiliate with the Twins to make up for Wichita moving up and one to become Nashville's AA team.
And that's not even accounting for the other expansion team – presumably Montreal – which would also need a set of affiliates; I could see them putting their AAA team in Ottawa, as the stadium is big enough and it would curry favor with the federal government. But that would displace (or adopt) a Frontier League team. So there's be more knock on.
And that's a big part of the reason MLB expansion is so complicated. Any MLB expansion inherently entails minor league expansion. So it's not just two new teams being created, it's ten.
But, as an aside, I don't think Nashville would be called the Volunteers. Sure, the name may have been used in the past, but that was a different time when the college sports programs weren't national powerhouses. I'd actually expect them to keep the Sounds name. I think part of the reason the AAA team changed their logo a few years back into something more staid looking was to set that up.