OP: I did Classics at Oxford, then a PhD, eventually got a permanent academic job.
After an MA and PhD, your DS is then looking ar several years on fixed term contracts. If he is lucky some of these may be something good like a JRF, post-doc, which lasts 2 or 3 years. If he is unlucky some of this will take the form of short term teaching only contracts, heavy load and poorly paid. There are many such jobs around in Oxbridge- many of the people who get them end up in these have no time to publish and so become uncompetitive for other jobs. Even if he's brilliant, there will be a number of years in this stage, when he will need to publish as much as he can, probably while being paid peanuts and with a heavy workload.
To get a permanent job in classics, he will need to have published his PhD as a monograph. Some people do still get jobs without this, but it's increasingly rare. He'll also need several years teaching experience, some articles, and probably some other stuff like involvement in public engagement activities, admin tasks for a department, etc. Even with all this, he will have to wait till a job comes up in his specific area of expertise (not often), and be geographically very flexible. And however good you are there is still a massive element of luck in the job market. I know brilliant scholars who didn't get a permanent job and other much more mediocre people who happened to be in the right place at the right time.
Overall, getting that first lecturer level job is tremendously stressful. If I had known what it would be like I wouldn't have put myself through it, and I am one of the lucky ones. Your son should talk to his tutors but ask them to be realistic (older Oxford dons may have got their job back in the day when you walked into it straight from a PhD).
That being said, lots of people do further study in Classics and then move on elsewhere - if he is good enough to get funding for an MA place, he doesn't need to decide until much later on.
Happy to answer further questions.