Setup
Why the limits. The trick finishes on the middle card of three, so three is the smallest row worth exploring. Above nine the chart stops being readable, and the search for every possible future grows sharply, since it has to consider every set of positions the card could be on. The signed card has to start on one of the cards, so that range follows whatever you set above. More than twelve moves only makes a very tall chart: the mathematics does not change, because parity repeats every two moves. Anything typed outside a range is pulled back to the nearest end.
Phase 1
The scrawl marks where the signed card could be. The arrows under it are the moves open to the spectator from there — at either end of the row there is only one.
Every path at once
Each row is one move, and a dot is a place the signed card could have got to by then. Notice the checkerboard: the dots never fall on two rows running, because a move always changes the position by one. The row you are on is highlighted — click any row to jump to it.
Take away a different card
I wanted to explore what would happen if you didn't just remove cards from the end — or even if you wanted to devise something that had a known multi-card out. For example, the signed card could end up with another card which is its mirror in the deck. I'm honestly not sure this does anything more than complicating things — but it might prove useful!
Phase 1 has already run, so the signed card is somewhere in the shaded cards below. Click any card to take it away, then make a move, and keep going. The two buttons decide what happens to the space: whether the remaining cards slide together and the positions renumber, or the gap stays open as a position the card can still be slid into. Nothing stops you taking several cards before moving — the trick alternates, one card then one move, but you may want two trees out of the way at once.
The number on the face is the card; the small number in the corner is the position it now occupies. They part company as soon as a gap is closed.
Every path at once
Same chart as above, built as you go. Columns are positions, not cards — so when a gap closes, the cards to its right shift into different columns. Below the line, hollow dots show every place the card could still get to, whichever cards you choose to take away next, in any order and with any number of moves in between: +1 is after one card has gone, +2 after two, down to three cards.