Many players would probably agree with me, that it's quite random on last stone(s). When players play well, it's mostly about good luck. This one change would make it more strategic and it could make tournaments more about skill and less about luck (as the game is otherwise more strategic than the last stones).
So my idea is:
Why not let players take 1, 2 or 3 times during the game a movement token instead of actually moving. When player takes a token, their turn is skipped (no extra turns) and they get a token with movement that they threw. E.g. if you get 5 from sticks, you could take value 5 movement token and end your turn. Later when you need it (e.g. when you are in House of Beauty) you could spend that 5 movement token. Meanwhile this token taking could also save player from horrible move they don't want to take. So there would be double benefit, but as the tokens are limited (to one or two per game) you have to be mindful when is the right time to take the token and when to use it.
With this change, the end game would be more about skill and less about which sticks do you get.