Great Britain beat New Zealand 3-1 to earn the final spot on the podium but White is still thinking what might have been had they not lost narrowly to Argentina in the semi-finals.
“It was heart-breaking to lose the semi-final because the gold medal was achievable," she said.
"I strongly believe we were better than Argentina in that game. We couldn't have done any more but decisions don't go in our favour and there's not much you can do about that.
"When you lose the semi final, you readjust your goals – I’m very happy to have won a bronze medal – we deserved a medal, it’s been a long programme to get here.
"I hope I’m still involved in Rio. I really, really want a gold medal. We’ve managed to hold onto our funding so I’m sure we can go for the title next time.
"We've beaten both Argentina and The Netherlands (eventual gold medallists) before. They're running scared of us because they know what we can do."