England narrowly missed out on reaching the Cup quarter-finals as they hosted the latest World Rugby Sevens Series event in London.
England needed to beat New Zealand by at least 10 points in Saturday's final game but had to settle for a 17-12 win.
That meant New Zealand and Ireland progressed from Pool C for Sunday's Cup knockout stage at Twickenham.
It also saw New Zealand join the United States and Fiji in clinching qualification for the 2020 Olympics.
There is still one place available via the World Rugby Series, with the London Sevens the penultimate event of the 2018-19 series.
But South Africa are ahead of England in the overall standings in fourth and won all three games on Saturday to reach the Cup quarter-finals as Pool A winners.
If England fail to finish in the top four they can still earn Team GB a place at Tokyo 2020 via a European qualifying tournament in July.
The hosts suffered a losing start to the day as they were upset 21-17 by Ireland, who were invited to take part before becoming a core team for the 2019-20 series.
England won their next two games to finish level on points with New Zealand and Ireland while Scotland, who are 10th in the overall standings, were bottom of Pool C after losing all three games.
Wales are 14th and beat Spain 21-17 to avoid finishing bottom of Pool D. The team that finishes 15th - the lowest core team - will be relegated for 2019-20.
Japan are currently 15th and they face Wales in Sunday's Challenge Trophy quarter-finals, while England play Samoa and Scotland take on Kenya. Ireland face three-time series winners Fiji in the Cup quarter-finals.