How to make fullscreen app's space appear next to the current space?

Clicking the fullscreen icon

while pressing the alt key will put it right next to the current desktop, at least on Mavericks.

I assume you mean that, when you make an app window enter fullscreen mode, you want that window to appear next to the current Space (by default it appears at the end of the Spaces list, as you say).

On macOS 11 Big Sur (and possibly older versions), you can do this as follows:

1. Drag the (non-fullscreen) window you want to make fullscreen up to the top of the screen.
2. Mission Control appears. Keep holding the mouse button.
3. The window you're dragging turns into a little thumbnail.
4. In the row of Spaces at the top, drag the thumbnail to the left/right of the Space where you want it to appear. A new space will open up, named after the app you're dragging.
5. Let go of the mouse button to drop the app into that Space in fullscreen mode.

(If you mean, how do you move an existing fullscreen app so it's next to another space, then just go into Mission Control and drag the app's thumbnail in the row of Spaces at the top.)