I have a few caveats:
Thanks in advance for your recommendations :)
Three books on antiquity that I have found to be of great quality and encompass a wide geographic area are:
Babylon: Mesopotamia and the Birth of Civilization by Paul Kriwaczek - this book is a great history that focuses not just on Babylon but the entire Mesopotamian/Near East region. It covers the creation of the first cities like Eridu to the Persian take-over of Babylon by Cyrus.
1177 B.C. The Year Civilization Collapsed by Eric H. Cline - I can never recommend this book enough to people. It is written by one of the world's top scholars on the Late Bronze Age in a way that's easily accessible to lay readers. The geographic area covered by Cline covers places like Mycenae to Anatolia, and the Near East. Cline does an amazing job illustrating the connections between Bronze Age civilizations in this book.
Ghost On The Throne by James Romm- This book is primarily a narrative history covering the years following Alexander The Great's death. It heavily focuses on the Macedonian elite but also includes how their power struggles influenced regions like Afghanistan and India.
I hope this helped!