There are about 4 hostels in Tartu Estonia that have their own local way of looking after guests. But Hostel Teviseks is new and opened by Americans.

www.hostelterviseks.blogspot.com

Between Tartu and the Latvian border there is the town of Voru. www.visitvoru.ee . If you want to stay somewhere stop in at Olle 17 the pub on Juri Street number 17 and speak to Bob or Ute the owners from the UK. They will probably find you somewhere to stay.

If you can get to Aluksne on the Latvian side you will be able to get a bus to Riga. There is one main bar in Aluksne www.aluksne.lv