I'm from there. June will be cooler(note i didnt say Cool), first of Sept will be Summer Hot.
I posted based on the US Climate Data website which states average temps are cooler in September. Problem with mean averages is they are not necessarily representative of reality and time of the month does matter.
As far as COVID goes yes, TN doesn't have any travel restrictions at the moment, which surprised me, but I think you are correct that NY may still impose some at that time. There may or may not be a high risk of cancelled plans but it's still something to consider. As we are heading to AK early June we are hoping much has been eased by then.