EDIT: for more information, check the new graphics right-click menu features guide.

Not sure how useful it is due to certain limitations, but I've just found (I haven't seen anyone doing it; sorry if this was already known!) that it's actually possible to keep the cursor in certain places during screen transitions. As far as I know, it only works when playing with the new graphics.

The trick itself is simple: position the cursor on an object, right-click and hold the button as Guybrush enters another room. Here's a video showing how this trick works.

There are some limitations, though.

First, it doesn't work when using the classic graphics, so it might only be useful for the Special% category.

Second, you can't open the menu anywhere; you have to right-click on an object. That means that the chances of leaving the cursor at the optimal place are pretty low. But in some screens it might position the cursor closer to the place you want to interact with. And there are actually places where you can get a perfect position. For example, you can position the cursor on the right when you enter the gaping hole on Phatt Island, and thanks to that you'll be able to quickly exit it.

Third, you can't hold period to skip lines of dialog while holding the right button, so it might not be very helpful on screens where characters are talking all the time, like when leaving Stan's.

Interestingly enough, opening the inventory after you've right-clicked on an object to open the actions menu allows you to stop holding the right button. The actions menu will remain open, so the trick will still work.