How do I manually update to the newest Spigot version?

You do everthing except update the WebUI commit in the WebUI.

The current commit version is shown in the “user-menu” in the top corner. (where current username (usually “mc”) and a small arrow is located).

There is also a fresh guide made by @hexparrot that goes throug a lot of the console commands:

(this includes links to posts about updating the WebUI.