Firstly unless there is user updatable content, is it actually a problem if for a few hours people access the old server?
Secondly you can configure your new server to respond to hostnames www2 as well as www, and point www2 to the new server FIRST - nobody will be using that name. Then when you set the redirect up redirect to www2.yourdomain.com or whatever.
Thirdly it shouldn't take hours if you do it properly: You need to use the TTL to solve this problem. Ask your DNS or hosting provider how, but it goes like this.
A few days before the move, cut the TTL to 30 minutes. On the day of the move, cut the TTL to 5 minutes. Then you get at most 5 minutes crossover.
Once the new server is happy, put the TTL back up to 48 hours.
Cheers, Ben.