If you are looking for accommodation options other than student dorms or real estate agencies, there are a couple of other options that will interest you.
Try getting in touch with other international students. Some of them have already found accommodation and might be looking for others to share with. Here are some international students and foreigner pages:
Here you can directly contact local owners renting their rooms/ apartments. Most of the posts are written in Estonian, so it is advisable to have your tutor or some Estonian friend help you write your post in this group.