When starting a new run, you are able to move while the level splash screen is still active, allowing you to do some inputs before you can see the game itself (this is also why the runs are timed by when the level loads, rather than your first input, cuz without being able to see sonic, it's pretty hard to judge that).

The setup that gets you this peel-out the fastest and most consistently is pretty difficult. If you press the "play" button on the main menu fast enough, you will be able to hear the short menu theme cut out as the scene loads, this is the earliest point that you can do an input that the game can read. By holding the peel-out button from that point until the charging sound effect reaches it's highest pitch, you will get the highest level peel-out as early as possible.

However, using this setup does mean that you'll be moving pretty fast towards wall and for about half a second without being able to see where it is. To stop yourself from running into the wall, you have to steer sonic slightly right while the screen is still black.