Well after looking at the link, it’s obvious that my illustrated method would be useless for the O.P.'s situation.
With a layout so limited for space, your staging will likely be from the shelf where you store the cars and locomotives not currently on the layout.
Take those from the layout which are going “elsewhere” (off-layout), and put the replacements “just arriving” from “elsewhere” (their boxes on the shelf) onto the layout.
The method mentioned above is the condensed method that my larger staging yards also use:
When a train enters one of those yards, it’s considered to have reached its destination, so all of the cars are remove from the yard, and returned to their respective boxes under the layout. The locomotive(s) then go to the nearest place where they can be turned (on a turntable or wye), and then return to their last staging yard location to have a new train of cars added to them, those cars heading to a different destination, either on-layout, or to another staging yard.
I can, of course, operate those staging yards as switching layouts, but it’s not really something which is of interest for me. I do, however, “switch” all of the towns through which these trains pass, as they travel from one staging yard to the next, so each industry is often receiving new loads (or empties) every time a train rolls into town, and cars are picked up, too, to be moved on, to another town, or to “elsewhere” (staging, then back in their box).
If you don’t have room for staging tracks, staging from wherever those cars are stored when not on-layout is probably your only option, if the carfloat idea doesn’t appeal to you.
Wayne