Altering TTL won't do anything for you unless you do it before you start to make changes and far enough in advance that the current TTL will expire and servers will pick up the new TTL. Then when you alter DNS records the alterations should propagate much faster.

If you're literally just changing DNS providers why would there be any down time - the record showing the IP where to find your website just gets grabbed from a different location, if a stale record is used it's still right.