Eric
Once the hosting change hits the DNS servers and propagates around the web, you are set. Until then, the old files should be available - just make sure the bill is paid with your old host so they don't pull the plug on you.
One month overlap is more than enough time. At worst the DNS change should propagate world-wide within a few days after you authorize your new host to make the DNS change.
Things can get complicated (although that's not common) - so make friends with tech support at your new host to ensure you take the right steps so that all goes as smoothly as possible.