Points to consider:
The existing Quick Start uses the term "Game Referee" — and in one case, an inadvertent "Game Master" — where the base game uses just "Referee." Think about whether the new Quick Start should be consistent with the base game. Besides consistency, "Referee" will simplify layout very slightly; longer paragraphs tend to cause more layout problems than shorter paragraphs.
The existing QS has a few instances of "mage" that should change to "magic-user." And its uses of "magic-user" should always include the hyphen.
Consider updating the spell descriptions to match the current version of the base game. Hold Portal refers to magic resistance, which is not defined in the QS (and could maybe be confusing to someone trying to learn via the QS), so either eliminate that reference, or add a note like, "The base game has details on magic resistance."
S&W has been sloppy about the definition of "an adventure." In the original game text, the term "adventure" means a _foray_ into a dungeon, or wilderness, or some other place. The "adventure" is not the _site_, but just one _expedition_ into that site, kinda/sort of matching the real-world use. The earliest pre-made sites were called "levels," "dungeons," or "modules." Around 1980, TSR started including the word "adventure" on the cover, as a descriptor of the content, but left "module" on the cover too. Eventually they dropped "module." S&W continues the practice of calling a pre-made site an "adventure," so it's consistent with the "modern" use. While I much prefer _other_ terms for pre-made sites, at least the Quick Start is consistent with the base game, so that's not horrible. However...
The QS uses the term "adventure" with its product-specific meaning long before it explains the nuances of the product specific meaning:
"The Game Referee Quick Start presents a very simple adventure for you to play, as well as some pointers and guidance on how a Game Referee can run a Swords & Wizardry session." (Does this imply the "game" is more like a play?)
"you get to pick if you want to be a Magic User or a Fighter for this adventure." (This is in the character creation session; the notion of "this adventure" is pretty hazy here; does it mean the must operate as the same class every time it goes into the Dungeon of Akban, even if he goes into other dungeons in between forays into Akban?)
"Playing Swords and Wizardry is very simple! You act as the characters in an adventure that you and the Game Referee create together." (Wait, I thought the adventure was the pre-made thing; but this says you create the adventure. What's the deal?)
"the players will do something unexpected during the adventure." (This implies the adventure is a time-based entity, matching the common definition of "adventure.")
Scrutinize all the QS's uses of "adventure," and look for ones to remove/change. While it's fine to use "Adventure" to mean the pre-made dungeon, take care to avoid excess confusion.