Remember that Wakanda has wanted to capture Klaue for a long time, meaning they never stopped looking for him. That's why we have this exchange (emphasis mine)
SHURI: You will never guess who just popped up on our radar.
A misidentified Wakandan artifact... was stolen yesterday from a British museum. We have learned Ulysses Klaue... plans to sell the vibranium
to an American buyer... in South Korea...tomorrow night.
T'CHALLA: Klaue has escaped our pursuits for almost 30 years. Not capturing him is, perhaps, my father's greatest regret. I wish to bring Klaue back here to stand trial.
Klaue just had a narrow escape from them, meaning he has to lay low or the Wakandans will send more people after him. As it turns out (pretty hefty spoiler)
Erik Killmonger is really a Wakandan prince. He only rescued Klaue from T'Challa so he could kill him himself, turn him in for favor among the local Wakandans, and then use that to foment a rebellion so he could take over Wakanda