There are many synonyms for the phrase "get after," which is typically used to describe pursuing or chasing something or someone. Some common synonyms include "chase," "follow," "pursue," "track," and "hound." Other synonyms might include "trail," "tack," "dog," "pester," or "press." Each of these words can be used in a variety of contexts, from chasing after a criminal to following up on a missed deadline at work. Ultimately, the synonym you choose will depend on the specific situation and the tone you want to convey, whether it's urgency, persistence, or determination.