Disk Space Upgrade

Some time ago, I increase the disk space on my virtual private server from 40G to 80G, but it's not reflecting on the server itself. I was told that I had to expand the partition. In order to do that, I have to back up my data. I cannot do that because the server is full. Can I order a one-time support ticket and allow them to do it? What other options are available for me so I don't lose my data?

The slow method.

Temporarily close huge sites on your server. Download the file content of these huge sites through FTP (CuteFTP client offers segmented downloads if you got very huge files). Delete those files from the server.
Now you have space.

Do what you needed to do with partitioning, then reupload the files.

I now have more disk space on my server by creating a cron job to delete clogged files. I still need to allocate the upgraded disk space, but I don't want to repartition the root directory. Is it possible to create a new partition and then allocate the disk space to the new partition? I tried using the resize method and it didn't work for me.