Unfortunately for A'Court, injuries began to take their toll. He missed the whole of the 1963–64 championship winning season, so Shankly signed Peter Thompson from Preston North End. After spending most of his career at Liverpool, playing 382 times and scoring 63 goals, A'Court was allowed to leave. He joined Tranmere Rovers for a fee of £4,500 in October 1964. A'Court's final outing in a Red shirt was in a historic match: Liverpool's first ever European match at Anfield, the game was a European Cup preliminary round 2nd leg match on 14 September 1964. Liverpool had won the first encounter with Icelandic side KR 5–0 but rather than treat the return leg as a mere formality, the Reds finished off the job with a comprehensive 6–1 victory.