make (out) after (someone or something)
To begin chasing someone or something. I made after the bus, but there was no way I could catch it. Tom made out after the thief, chasing him down an alleyway around the corner from the building.
make (out) after someone or something
to run after someone or something; to start out after someone or something. Paul made out after Fred, who had taken Paul's hat. The police officer made after the robber.