これはもう丁寧語で、普通、目上の人に対して、「あなた」さえも言わないだろう。 例えば、「あなたの帽子ですか」 この帽子は先生の、あるいは目上のだったら 御帽子ですか　となるだろう。

Just as an aside; I also was told to avoid anata for the same reason and use their name-san instead. What I've found when I converse with a Japanese person, is that when I want to change the topic and begin a sentence addressing the person by name + particle wa, the Japanese person will often say, "hai" in response to hearing their name. So when I then complete my sentence, perhaps a question (seeing as I've just addressed the person by name) it can make the conversation seem more natural.

Basically "his name-san". It's not odd but common.

So, if I needed to mention the person I'm talking (with), instead of "anata", can I say "his name-san" ? Or it will be odd?

In language school, teachers teach you "you should not say Anata" because if you don't know the proper situation, you might make someone to be uncomfortable. To repeat calling someone "Anata" could be impolite. An advanced people will gradually know it when is okay, when is not.