The river Styx, like most of the Underworld, follows no consistent, constant geography. Instead, it's in a constant state of flow, shifting and changing its banks and course as the realm itself rearranges its structure.
That is only one of the many ways the realm has of keeping its reputation as inescapable, Lord Hades's rulings being the only thing that can grant one permission to leave. This has happened only a handful of times in history, all of them exceptional in one way or another. And as far as the living and the vast majority of the dead are aware, this is an absolute fact of the world, as unchangeable as gravity.
This is not to get into the fact that, unless you were sent to the bowels of Tartarus for punishment, you would have very little reason to want to escape. There was a brief period during which the Asphodel Meadows were flooded with a burning river a few centuries ago, but the Queen of the Underworld has long since dealt with that and the land of the dead is actually a reasonably pleasant place to spend eternity in.
Very few people are aware of the one exception to the rule, taking the form of the Prince of the Underworld, Lord Zagreus. He has been the first and only person to ever escape against his Lord Father's wishes, and although he cannot stay on the surface for very long due to an intrinsic conflict between his godhead and his mortal blood, he has become proficient enough at it that it was made into a job, finding security flaws and other issues that might jeopardize the so-called inescapability of the realm.
Now one of the few fixed structures of the Styx, the small pool of it which ends right in front of the hall at the House of Hades, sees its surface bubbling once more before the god of life's figure emerges from it, blinking some crimson liquid away and cracking his joints to get rid of the phantom pain from his most recent death. The water slides off his body and clothes with preternatural haste as he walks up the steps and past the long line of shades standing before his father's throne and desk to petition for one thing or another.
"Hello once again, my friend," he says, nodding a greeting in the direction of Hypnos, god of sleep and official secretary of the Underworld.