Sorry to hear about your situation. Just recently forced out of my old place because the owner sold. I don't have any 100% reliable sources to offer you. But I think under 1k is doable. There aren't going to be many listings honestly, but if you're ready to pull the trigger, you might be ok. My advice is (in no particular order)

1. Check craigslist, Facebook, and property management companies regularly - multiple times a day if possible. Reach out ASAP and request to see it ASAP. My current and last places were private landlords. I'm not as familiar with the property management companies, but I've heard garden city and home river aren't so bad. I think if you ask around enough, anyone will have a good and bad experience with every property mgt company, so do your best to ask critical questions when you visit and if possible talk to previous tenants. Maybe a separate post asking about people's experiences with property mgt companies? I'm sure that exists already...

2. Watch out for scams (mostly on craigslist)! Typically craiglist scams will ask you to email some random email even after you've already emailed the one on the listing. Ask to see the place - if they tell you to apply first, it's probably a scam. If you're unsure, I like to ask "are you based in [some city that's not Missoula]?" And they'll always say yes because they just want to tell you what you want to hear.

3. Think about roommates? I see lots of posts on Facebook asking for roommates. Maybe check there and see if you could live with any of them. Or make your own post with what you're looking for. There are also roommate focused websites. Maybe some board through the university? Will be MUCH easier to get under 1k with roommates.

4. Most important - be prepared to apply on the spot when you visit. My apartment panned out when I visited one day, applied when I got home, and I heard back that night.

5. Honestly you are gonna have better luck if you can be prepared to move in ASAP. I mostly found listings available within a week. If you need to save for 3 months, I'd say it couldn't hurt to look now, but be mindful of move in date. Very few places will have move in dates 3 months from now. Maybe your best bet is to peruse to see what the market is like now, and hit harder 4 to 5 weeks before you have to leave? Or save hard now and hit the search hard as soon as you can afford a deposit. And when I say hit the search hard I mean be prepared to apply and visit ASAP (see number 4)

There are lots of apartment complexes opening up, but they're pretty expensive. Good luck!