head out after
head out after (someone or something)
To trail or pursue someone or something. You better head out after the guide if you don't want to get lost. Fearing the police wouldn’t get there in time, the store clerk headed out after the robbers on foot.
head out after (someone, something, or an animal)
to start pursuing someone, something, or an animal. The sheriff and his men headed out after the bank robbers. We headed out after the runaway boat.