Once you have FTP details to the old hosting account, download a copy of the original site, or if it involves a database etc., do a backup of the original site using something lie CPanel (if the old host has a control panel). Once you have FTP details of the old host, you could also just transfer the whole site to the new hosting account through that hosting account's control panel, or do the same thing via your command line (you can google all that stuff).
While building the new site, you can leave the old on up on the new hosting until you are ready to launch the new version of the site. You can either build the new version on your local computer, or perhaps at a subdomain of the site (such as staging.clientsite.com).
Anyhow, once you've set up the old site at the new hosting, just redirect the domain name to the new hosting account.
Hope that makes some kind of sense. I'm still not clear on the workflow you are picturing. Do you want the old site to stay up until the new one is ready?
Another option is to leave the old site in place, while building the new site at the new hosting account. You can reference the new site at its IP address, and then when it's finished, redirect the domain to the new version of the site. So there are lots of options.