You can use either 'that' or 'who' in this sentence, because it's a defining relative clause. There's no difference at all in meaning. You could go for 'who' in this case to avoid the repetition of 'that', but it really doesn't matter. Either one is fine, and 'that' tends to be used more in speech because it fits more easily into spoken sentences.
The only situation where you do have to use 'who' is when it's a non-defining relative clause. In other words, if you can omit the relative clause without affecting the structure and meaning of the sentence, you have to use 'who' for a person.
Let me know if you need any further explanation, or if your 'gut feeling' can take over at this point!