I used to go to this place pre-pandemic up in north Austin and they do. I just started shaving my own head so I no longer need a barber but they were great. They appear to have more south locations: https://finleysbarbershop.com

Rosewood Barber Shop offers the most quality cut in town IMO, and they prefer online booking: https://www.rosewoodbarbershop.com/

I have been going to McGuire’s on Brodie since they separated from The Nines. Pricey, but very worth it, especially if you like a quality beard trim.

South side is lacking in good barber shops as far as I have seen. I go to Little Barber near the university and they’ve never done me wrong. Their online booking is a standard experience.

Floyd’s on Brodie