If your mini supports Sequoia it should get all updates. You can also download the full installer to update, it's in the AppStore or use the direct link (faster). Both links are in the first post here.
This is the 15.7 (Build 24G207) Release Candidate full installer. You can skip 15.6, there is no letter at the end of the build number, so it's not really a beta anymore and could be identical with the final version of 15.7.
https://swcdn.apple.com/content/dow...mxpzbcx08l13u0f59im4s5ph/InstallAssistant.pkg (15.65 GB)
You can also use the Teminal:
softwareupdate --fetch-full-installer --full-installer-version 15.6
or
softwareupdate --fetch-full-installer --full-installer-version 15.7
Edit:
Do
softwareupdate --list-full-installers
before, to see what's available. You should at least see 15.6 there. The newer ones might only appear if Developer Updates are activated.
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