Daegu is a third biggest city in South Korea, quite a pleasant place and the one which is actually very easy to hitchhike out of.
It's easy! Just take the red subway line (there are only two in Daegu) to the Yonggye station on the east part of the city. Leave the station at exit 2 and turn left. After about 300 meters you will see the highway with the big Dongdaegu (east Daegu) intersection and the toll gate. Just walk past the gate and there is plenty of space to put your thumb out. (You have to walk along the road, which can be dangerous. But a alternative route on the west side of the tollgate exists. Just check kakao map.) Don't forget to make a sign in hangeul, because the highway splits in two close to here: highway nr.1 going via Gyeongju, Ulsan to Busan) and the nr.55 one directly to Busan.
Update 05.28.2017: The staff at the tollgate asked me to leave. But this could have been a random thing.
Update summer 2018. After reading the comment above that the staff kicked the previous hitchhiker, I still wanted to try this spot to go to Gyeongju. I walked to the tollgate and immediately the staff came out and angrily told me to get off, that I could not hitchhike there etc. I tried arguing politely with them to let me stay there and that I could quickly catch a ride (the spot was perfect) but they threatened to call the cops on me and were very aggressive and angry because I was resisting their BS (every other tollgate in Korea let me hitchhike and even helped me). They said it was because the station was part of the highway. I eventually had to leave and took a bus instead, it was the only instance in Korea where I could not hitchhike out of a city. Here is my advice for you if you need to hitchhike east from there: instead of walking all the way to the tollgate, stand on the side of the road leading to it and make yourself very visible. High chance that a car will stop to pick you up and the danger is limited because they don't drive very fast since they need to stop at the toll. The tollgate staff likely won't notice you because it is far from the tollgate itself.