The fairness claim also caps the raise at what the last person got and tips your hand in terms of information that the counter party to the negotiation does not know that you have: the counter party does not know that you know what other people are paid.
My random internet advice is to have an independent case for a raise based on your work. It can be helpful to describe some specific financial objectives inside the request, for example "I am hoping to buy a house and the raise will help with the down payment" makes the request seem more concrete and safer (e.g. you did not say you're going to spend it on illegal drusgs). It can also help the counter party think about the benefits of providing the raise and feel better about providing it. Just like few people want to be accused of being unfair, many people want feel better if they help someone "buy a house".
But again, focus on your work. The advantage you have is that you can point to what you accomplished for the business whereas new hires cannot.
Good luck.
If the new hires told you their salaries the difficulty is discussing it without revealing this fact. Management sometimes frowns on that.
If the pay range was publicly disclosed in an advert that would be the best way to go.
In general however it seems like an entirely reasonable thing to do. Just don't get too aggressive.
So don't undervalue yourself, you can ask for nothing and if you don't get it just move on to a place that values you.