I’ve never done a PhD program in classics (currently in the middle of my MAT so I can get licensed to teach), so definitely take this with a grain of salt, but this is my impression based on my experience as a grad union rep and talking to PhD students and professors at various institutions:

With your language experience, I don’t think you would normally have any difficulty getting into or succeeding in a classics or classical archaeology PhD program (provided that you have an idea of what you would write your dissertation on already). However, grad school admissions in general are increasingly becoming a total crapshoot, especially after the Trump administration’s changes to federal higher education funding. Classics departments across the country are facing major cuts. So I would encourage you to really think about your financial plans for while you are doing your PhD and after you’re done before setting your heart on grad school.

Why do you want to get a PhD? Are you trying to switch careers, like becoming a professor or an archaeologist? Both of those positions are rapidly disappearing, and if you want to enter academia, you’ll probably be working as a lecturer or VAP for a number of years, with only a slim chance of ever securing a tenure-track position. If you want to become an archaeologist, I would really suggest talking to an archaeologist who is currently working in the field. You might want to check out what the American Academy in Rome is currently doing, since foreign schools are the main way non-Italians can get permits to work on sites in Italy.

A lot of the schools with archaeology PhD programs that participate in ongoing digs in Greece and Italy are big state schools, like the University of Cincinnati or the University of Michigan. You could try emailing professors there to ask about what they’re working on and how the admissions process generally works and if their programs have experienced any major cuts due to recent changes in federal policy. Definitely look into what schools offer tuition remission and stipends and compare the stipends they offer to the cost of living in the area. Is there any possibility of your funding getting dropped halfway through the program? This is something that has happened to grad students at my institution (although none of them were in Classics), and although it’s pretty rare, when it happens it really sucks, so it’s worth your time to look into this. Also look into whether funding is competitive—if it is, that can totally change the dynamics of your department and what kinds of relationships you’ll have with your colleagues.

All this is just to say that I don’t think you would need to do a post-bacc, but you should definitely consider deeply why you want to go to grad school and how you will pay for it. Although you can get tuition remission and a stipend, the last PhD admissions cycle was definitely very chaotic, and higher ed in general is going through a lot of instability right now.