But all that could change with their next three Tests on home soil and at altitude.
"It was a huge character test for every player and for every member of the management team," White told Beeld about the last five weeks, which he described as the toughest in his three years in charge.
"The tour began on the worst note imaginable with a record 49-0 whitewash against Australia in Brisbane. It would have been a knockout blow to many other teams.
"But we got up and, a week later, played much, much better against the All Blacks. Only small mistakes kept us from beating the best team in the world.
"On Saturday, we already had victory in our grasp before it was snatched away four minutes before the end of the match.
"It was a laudable performance by the Springboks; one that every player can be proud of. The players really wanted to finish the tour on a winning note because they had suffered so much criticism.
"I appreciate that one can never be satisfied with a defeat, but on Saturday the Springboks showed great character, courage and pride," White added.
"What is even more encouraging is that the players, despite a solid performance, were so disappointed about the result.
"Just as our supporters do, the players set high standards for themselves. Just as much as our supporters want to back a winning team, the Springboks want to play winning rugby."
White believes the performances will improve at home.
"At home, the Springboks have always been a greater factor to contend with," he said. "This is shown by the fact that we (since he became coach) have lost only once at home."
The Wallabies have not won in South Africa since 2000 while the All Blacks last won in the Republic in 2003.
The Boks face New Zealand twice, in Pretoria on August 26 and at Rustenburg on September 2.
The Wallabies are their final Tri Nations opponents at Ellis Park on September 9.