Your 2013 Mac Pro is able to run macOS 10.13 - 12.x if you mean the desktop "trashcan" model. Even a 2013 laptop can run macOS 10.13 - 10.15 or 11.x depending on the model.
If you don't want to disrupt an older OS on the internal drive of that 2013 Mac, then just install macOS 10.13+ to an external USB3 drive. Just make sure to have a good backup of the 2013 Mac first just to be completely safe. The later macOS 10.13+ installer will update the system firmware, but otherwise there should theoretically be no impact to the currently installed older OS on the internal drive. Boot the 2013 Mac from the macOS 10.13+ external drive and access your data so you can transfer it to another external drive.
You can find links & instructions for downloading various versions of macOS using this Apple article:
How to download and install macOS - Apple Support
FYI, you may first need to install macOS 10.13 to the external drive before being able to use later versions of macOS. Keep this in mind if you have trouble installing a later version of macOS first. You can also perform an online search for the system requirements for whatever version of macOS you want to try to install to the external drive. For example: macos catalina system requirements
While you should be able to install macOS by running the installer while booted from the internal SSD containing the older OS, you may want to consider creating & using a bootable macOS USB installer:
Create a bootable installer - Apple Support
Just make sure you first use Disk Utility to erase the external drive as GUID partition and MacOS Extended (Journaled) so it is ready to have macOS installed on it. Give it a unique name....maybe "Catalina HD" if you are installing macOS 10.15 Catalina to the external drive. Plus make sure to select the external drive as the destination for the macOS installation when you run the installer whether from the bootable USB stick or while booted to the older OS on the internal drive.
While you can use a hard drive for the external boot drive, it will run slowly compared to a USB3 SSD. This may be Ok since this is a temporary situation so you can transfer your files to the APFS external drive.