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.
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
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.
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.