Change the narration: "I’ve been working for the same company since 1960" he said to me.

He told me that he had been working for the same company since 1960.

He said to me that he have been working for the same company for 1960.

He said to me that he had been working for the same company for 1960.

None of these

He told me that he had been working for the same company since 1960.

Direct speech: "I’ve been working..." (present perfect continuous)

Indirect speech: changes to past perfect continuous → "he had been working..."

"Since 1960" stays the same because it refers to a specific point in time.