Will taking the ACT a third time be beneficial?

Have you taken the SAT? It might be at least as beneficial as re-taking the ACT. I think you should try the SAT at least once. Even if your composite isn’t much better, you might have better results on some sections. My son’s final results, after second rounds, were a 31 ACT and 2060 SAT. Those are very similar results, but his section scores varied, showing strengths in all areas (ie. 730 Math SAT/29 Math ACT; 620 Writing SAT, but either a 33 or 35 - I confuse his English and Writing scores - on ACT).