Transfer account will be best as it will fetch latest file/database updates. What I do is start a transfer of multiple accounts and watch the progress. When each account is trasfered and created successfuly on the new server, I immediately change dns so that it points to the new server. (you can either change name servers at the domain or change the ip-s in your old servers's dns server). You could also reduce one day beforehand the dns refresh of your current server, so that old ip-s dont get cached for a long time in dns servers around the internet.
I would suppose that this way is as quick as the backups method because they both involve the same operations like dups, gzipping, etc.