For that matter means what you said about one case is also true of a wider one. I haven't visited Rome, or any Italian city for that matter: not Rome, and not any other Italian city either. It usually goes at the end of the sentence after or or and. It sounds conversational, but it's fine for IELTS Speaking.
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statement + or / and + wider case + for that matter
- I haven't visited Rome, or any Italian city for that matter.
- The rule applies to managers and, for that matter, to volunteers too.
- She doesn't enjoy football, or team sports for that matter.
What changed
First versionI haven't visited Rome, or any Italian city for this matter.
RevisionI haven't visited Rome, or any Italian city for that matter.
You're talking about the topic right now, so this feels closer and more natural: for this matter. But the expression is frozen with that: for that matter. This was replaced with that.
As for moves to a different topic: As for the volunteers, they arrive at nine. For that matter keeps the same idea and makes it wider. Not to mention adds a new point on top: The flat is cheap, not to mention close to work.
Your turn
The same statement applies more widely: I don't eat beef, or any meat for ___ matter.
Show the answer
that — The expression is fixed: for that matter. This seems natural for the current topic, but the phrase isn't used with it.
Say what you don't do or don't like, and widen it to a whole category with for that matter.
He doesn't use social media, or a smartphone for that matter.
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