Searching for accommodation for the winter semester was rather difficult, since fewer UT students go abroad at this time of year, resulting in a higher proportion of incoming students. Also, most apartment buildings outside the campus only offer full-term leases. Thankfully, I found my accommodation through Josh Rucker (my academic advisor at UT), who sent an email to all exchange students about a business student subletting her room at Lark Austin for the Fall semester. I responded immediately, but due to visa issues, it took some time until I finally signed the lease (I signed in July, about four weeks before flying to the US).
Also, don't have too high expectations: the campus apartments are promoted as luxurious, which they definitely are to some extent, but there is less 'Mieterschutz' compared to Austria, and some odd fees (e.g. if you don't pay with an American card, there's an extra fee of about $100, but this can easily be avoided by creating a US Revolut account and paying with it).
Although it's not perfect, I would live there again as being that close to campus, friends, and the social scene in general was totally worth it. In general, I don't think there's such a thing as the perfect apartment if you're on a budget like most exchange students are! For example, my friends had issues with long waiting times for elevators, which I did not experience. Just do proper research, and if it sounds too good to be true, it probably is!
Before I found my apartment, I used platforms such as the UT student housing platform. There was also an Excel spreadsheet created by Josh Rucker where incoming and outgoing exchange students listed their apartments, and there are multiple Facebook groups. However, be aware of scammers. Since I had confirmed my apartment pretty early on, I stopped looking at some point, but it's a good idea to start early and regularly check for new apartments.
One thing I wish I had known beforehand is that, even though it's not listed as a housing option, you should try to apply for the dorms via the international student housing program of ISSS. Some of my friends lived in the dorms, and I would have loved to as well. Just apply as soon as you receive your confirmation from UT, and be open to sharing a room. I think I would have got in if I had selected "single room" as well as "double room". Also, open a US Revolut account as soon as you arrive to avoid the aforementioned card fees. If you need to transfer money to the account, use a different Revolut account (e.g. from your parents, not possible to have two at once) to save on transfer fees.
However, don't let this discourage you, it will be so worth it in the end, and I would choose UT again, even though the housing search was a bit complicated and also a little bit tied to luck in my case.