I have a long list of small complaints about Spotify and, until today, one of them was the lack of an option to re-order tracks in the playlists that I've created. I could re-arrange my playing queue after choosing a playlist, or go to the desktop or web player to edit that playlist, but on my Android phone, the "custom" sorting order wasn't modifiable. Now, it appears the function is rolling out.
We knew this was coming when Spotify said it was going to work on it a few weeks ago. Thankfully, that didn't take long. If I head over to any of personal playlists, I can tap the overflow button on the top right, choose Edit playlist, and then have the option to rename it or change the order of all tracks.
Once done, I hit Save, and the new order becomes the default custom one.
Spotify likes to tinker with server-side updates a lot, so it's possible this feature isn't live for you yet or that it's been live for you for a while. I'm on the latest stable v 188.8.131.523 of the app (APK Mirror), and both David and I can see the track re-ordering.