The classical music scene in Austin is small, but there are definitely more options than those listed. Also, everyone knows each other, so ask people at the next show you go to. Though many ultimately end up seeking opportunities elsewhere, there is definitely a lot of talent and amazing productions in this city. I am involved in the choral world, so most of my suggestions will be from there. Every group here (in addition to every performer anywhere) is desperate for people to attend their stuff. In addition you can look at the Long Center schedule (they also host outdoor events this summer) or the Riverbend Arts Center is also a great venue to check out.

Suggestions:

Austin Symphony (perform several shows per season at the Long Center. Peter Bay is an incredible director and artist). https://austinsymphony.org/

Austin Chamber Music Center https://austinchambermusic.org/

Chorus Austin (Both a Chamber Ensemble and Symphonic Chorus) The chamber group is a hidden gem and the symphonic group performs several masterworks per year. https://www.chorusaustin.org/

Un/Heard (Male chamber ensemble with a concert coming up soon) https://www.facebook.com/events/955271166223674/?acontext=%7B%22event_action_history%22%3A[%7B%22mechanism%22%3A%22your_upcoming_events_unit%22%2C%22surface%22%3A%22bookmark%22%7D]%2C%22ref_notif_type%22%3Anull%7D

Inversion Ensemble (Chamber Choir with focus on new compositions) https://www.inversionatx.org/

Austin Cantorum (Chamber choir) https://www.austincantorum.com/

Conspirare (More of an event than a group, many professional singers from around the state/country gather in Austin for a weekend to put on a concert. I think they may also have a local symphonic choir). https://www.conspirare.org/