Heading due south from New York, one could pass through Pennsylvania, Maryland, West Virginia, Virginia, North Carolina and South Carolina.
Heading due south from Michigan, one could pass through Ohio, Kentucky, Tennessee Georgia and Florida OR Indiana, Kentucky, Tennessee, Alabama and Florida.
Heading due south from Pennsylvania, one could pass through West Virginia, Virginia, North Carolina, South Carolina, Georgia and Florida.
Of course, which states one passes through depends heavily on where in the first state one begins the trip and which route one chooses to follow.
Your question is a bit misleading. There is no US state that touches six others on its southern border. There are several states that touch up to eight other states, but there's no US state that borders six others to the south.