You are quite right that it seems counter-intuitive, and but it is the correct form.

In reported speech form the writer would have put : I said," I have time next week. He said that he had time the following week". Now the tense for the speaker is in the past and he is reporting what was said after his first statement at a time in the past. If it was ...I say" I have time next week" then it becomes "He says that he has time the following week".

Be careful to report the speech accurately. "He said (that) he has time next week." "I had time the following week." "He said (that) he had time the following week." The only thing that you said is the pronoun construction and verb agreement.