On Thursday, October 8, Thompson resigned his third game with Morphy after forty-six moves, and two hours and thirty-six minutes of play. Now Morphy advanced to the second section. Later that day he played side games with H. P. Montgomery, Louis Paulsen, and James Thompson, winning all games, the last given here in facsimile of Morphy’s writing.
Lawson then gives the game as recorded by Morphy himself. The reader can click the picture at the left and see an example of Morphy's neat handwriting. A modern politician could accuse Lawson of ignoring "the elephant in the room" when he declares this game played after the tournament game. Clearly, Morphy noted at the top of the score that this is game 3 of the CI or Congress I.

A better explanantion is that Shibut 125 was played before Shibut 003 which was published as the actual tournament game. Thompson played poorly and as Löwenthal noted:
The termination of this game is pretty, and it was solely on that account that we included it in our selection, as it otherwise presents few points of interest
Could it be that, given the poor quality of play and short nature of the game, Morphy allowed Thompson to play the third game again? Could the players have turned in their second game played that day (Shibut 003) as their tournament game? We may never know for sure but at the very least the exact date of Shibut 125 is known.
[Event "Unofficial Games From the 1st ACC"]
[Site "New York, USA"]
[Date "1857.10.08"]
[Round ""]
[White "Thompson, James"]
[Black "Morphy, Paul C"]
[Result "0-1"]
[ECO "C52"]
[Opening "Evans gambit"]
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.b4 Bxb4 5.c3 Ba5 6.d4 exd4 7.Qb3 Qf6 8.O-O d6 9.e5 dxe5 10.cxd4 exd4 11.Nbd2 Nge7 12.Bb2 O-O 13.Ne4 Qg6 14.Bd3 Bf5 15.Nh4 Qh5 16.Nxf5 Nxf5 17.f4 Rfe8 18.Rf3 Nb4 19.Ng5 Nxd3 20.Qxd3 Qg6 21.Rh3 Re3 22.Qb5 Rxh3 23.Qxb7 Re8 24.Nxh3 Bb6 25.Rf1 Ne3 26.f5 Qg4 27.Nf2 Qe2 28.Rb1 Nd1 0-1
Game Sources:
Lange - Game XIII
Löwenthal - Page 397
Maroczy - Game 054
Sergeant - CLXXVII
Shibut - Game 125
Date Source:
Lawson, page 58