Besides remembering the levels positions on the world map, there are some ways to speed up the transition between levels and load the next one faster:

• Press "space" right after picking any puzzle piece, to skip the menu and get to the world map faster.
• Keep the mouse on the position of the current level, and after the current level ends move the mouse to the next level (short distance move)
• Alternatively, right after entering a level, move the mouse quickly to the position of the next level, and it will be ready right when the current level ends. The drawback of this method is that you have slightly less time to position your hands on the keyboard before the level begins, potentially losing time there.

The games hides the mouse icon while inside a level, so it won't be a visual bother.