Sir Ben Ainslie's boat finished second in Saturday's two races in the America's Cup World Series in Gothenburg, Sweden.
The Land Rover BAR team finished runner-up to America's Cup holders Oracle Team USA in both races.
They had looked set to finish third in the second race but overhauled Emirates Team NZ late on to earn vital race points.
Sunday's schedule will see a further two races with double points available.
Ainslie's team won the opening leg of the series in the Solent last month.
This weekend's racing is part of a lengthy qualification process that will count towards the 2017 America's Cup Challenger Series, the winner of which will take on Oracle in the 2017 America's Cup.
It will be the 35th staging of the oldest competition in international sport which was first raced in 1851 around the Isle of Wight.
It was named after a boat called America, which beat the best Britain could offer. A British boat has never won the trophy.