Reported speech

I am not a native speaker. I would like to know if the following rule is correct or not :

Main Clauses connected with and / but

If two complete main clauses are connected with ‚and‘ or ‚but‘, put ‚that‘ after the conjunction.

Example:He said,“I saw her but she didn’t see me.“ – He said that he had seen her but that she hadn’t seen him.“ Is it correct to say : He said that he had seen her but she hadn’t seen him. I normally don't use "that" after the conjunction.