You may try for El Capitan as that would be a newer old
system ~ up from Mt Lion & conditionally still available.
This may likely be key to go further up the chain; I believe.
• How to upgrade to OS X El Capitan - Apple Support
https://support.apple.com/en-us/HT206886
But if you use the links from various macOS upgrades you should be able to get one
much newer. Problem has been with those who have older 32-bit software and drivers;
while the latest macOS 10.15 solely uses 64-bit. That would be Catalina. Mojave is tamer.
My Late 2012 Mac mini 2.3GHz quad core (server) last model shipped with Mavericks;
that's still on HDD#1 on partition #1; while El Capitan is on same drive, partition #2.
This does OK, after I realized a need to boot into second system requires different
keyboard shortcuts. Now I seldom revisit Mavericks; it has near 500GB space on its
partition. Same for El Capitan's space. (My Mac mini has two 1-TB 5400-RPM HDDs.)
So the deal is mostly this; there are systems above Mt Lion that are internet
downloads that can be used. But your original Mac OS X is what will load by
default when the system is restored. (Unless upgraded further, and a firmware
update may keep it from using original Mac OS X. )
The other keyboard commands suggested in with later installer download
page w/ support articles with links to get them, can yield options; and you
can make the computer you have upgrade to later mac OS.
• How to upgrade to macOS High Sierra - Apple Support
https://support.apple.com/en-us/HT208969
I'd downloaded High Sierra installer about three years ago; it lives
as such in the Applications folder until used to get the actual 10.13.
system from Apple Server; or when I choose later one. There are
some issues some users had with various hardware; so these are
best not taken to the newest version. Each significantly changed
behind the curtain; the latest three, much more.
If your Mac mini has second hard drive, that could have originally been
a server; the HDDs are identical if from Apple. Otherwise there was a kit
with extra bracket (and perhaps second data cable for 2nd HDD.) There
are a few work-arounds if you choose to run with older mac system;
but that may be problematic.
Good third-party 'browser' and 'email client' applications are available
from mozilla (firefox and thunderbird.) Safari is only as up-to-date as
the system is, so it can be rather outdated. A few years behind, if not
given separate update steps along with security updates from Apple.
You should re-familiarize yourself with specifications & hardware
upgrades that your older Mac may need for a later macOS.
Good luck & happy trails!🌞🌤