No one can ever, ever offer truly unlimited bandwidth or transfer. That is a marketing ploy where they limit you by CPU cycles or other ways so that there in fact is a finite end to the resources.
A real host, one that will do the full transferring of your current sites is
Amerinoc. They will do the transfer for you and has rock solid support, there are many reasons as to why they power my entire online presence. I made the switch many years ago and haven't ever had any reason to look back.
You too should make the move to
Amerinoc! 