Sure you can transfer the data from your old site to your new site.
For most content like HTML files and images, you can use any FTP client to get the files from one place and then upload them to the new place. Or use something like Beyond Compare to connect to both FTP servers at once and skip your own computer.
Some data, like databases and lists of e-mail addresses, may be more tricky. E-mail etc. is best done by hand so you know the exact setup of your new site. Databases is different, but perhaps you’re not using any?
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