Can I upgrade my Mac but not to High Sierra?
Since you've clarified that you've never downloaded an installer for a later version of macOS, then you've only got three options:


Download an installer from a 3rd party (not recommended)
Acquire an installer from a trusted friend or other trusted source
Try your luck at an Apple Store (or other authorised outlet) to see if you can find one willing to do the upgrade for you


There may be other workarounds open to you. For example, if you have access to another Mac (yours or a friend's) that is not capable of running macOS High Sierra, but is capable of running El Capitan or Sierra, then you could start your Mac mini in Target Disk Mode and set it as the default startup disk for the other Mac. Then you could use the optioncommandR startup keyboard combination to upgrade to the latest macOS that is compatible with that Mac. You'd have to make sure you select the Mac mini's hard drive as the install location.

Of course, I can't stress enough how important it is you have a backup regime in place, be it via Time Machine or some other solution.