Kuressaare is a small town in the island of Saaremaa, in Estonia. Visiting Kuressaare by hitchhiking is easy. From other towns in Estonia like Tallinn, Pärnu, Tartu etc. it is helpful to have a sign "SAAREMAA" with you to catch longer rides and not get stuck in small places on the way where cars wont stop due to high speeds.
First you need to get to Virtsu and catch a ferry. If you are already in a car also going to the ferry just hand the driver your ferry ticket money (2,24 € for adult) You can check the timetables and prices here. If you go on the ferry on foot it is important to run off it before the cars and put your thumb up, because missing a ride here means at least 40 minutes waiting until the next ferry. Other option is to ask from drivers before landing. Most cars go to Kuressaare and it takes less than an hour.
Towards mainland - the only spot to hitchhike back towards mainland is right after the roundabout north of Kuressaare. There is shopping centre Auriga closeby. You can get to the spot with an intercity bus nr 4 going to Kudjape, but it is only 2 km from the centre, so you can also just walk. Try to catch a ride which takes you at least until Kuivastu port, it is difficult to catch a ride in between.
Towards any other direction you also need to get to a roundabout and choose the right direction. Traffic might be quite low on local roads and it might be better to use public transportation, bus tickets shouldn't cost more than 2€. Easier places to hitchhike to are probably Mändjala, Salme, Kihelkonna, Aste.