Start with the full raw headers from the delivered message.
1. Build the timeline
Read trusted Received fields and record each timestamp. Work from systems you trust backward through the path.
2. Find the gap
If one hop accepted the message at 10:02 and the next did not receive it until 10:47, investigate the system responsible for that interval.
3. Correlate logs
Use queue IDs, Message-ID, envelope addresses and timestamps to find the transaction in SMTP or gateway logs. Look for 4xx responses, connection failures, DNS errors, rate limiting or TLS failures.
4. Separate queue delay from mailbox delay
If the final receiving service accepted the message promptly, later client synchronization or mailbox processing is a different problem.