"It was important to get a campaign like that behind us and lay this away record to rest with the World Cup and another Six Nations coming up," White, the Scotland skipper, said.
"There are a lot of games in the World Cup and we have got to learn how to play well away from home.
"We have taken a huge step but now we have to take the next step.
"Our away form has been improving and I hope this win in Italy is the first of many.
"We can use this as a springboard. Winning away from home has seen us overcome a big psychological barrier.
"Now we have shown we can compete and make our mark."
Scotland have three games at Murrayfield in the next Six Nations, and with the Scottish public behind them again, the back-row forward claims they will be major contenders.
White added: "The last couple of years have been hard for us, so to finish third in the championship was a great achievement and a step forward.
"It gives us something to build on and we are looking to get more victories in the autumn games first.
"In the next Six Nations we have three games at home and we will get big support after what we have done this time.
"We have got the nation behind us again and now we have conquered our fears away, we can make a real impact.
"I was really proud to captain my country in the Six Nations for the first time. This sends out a message to the rest of the teams that we have turned the corner."
The Sale Sharks man believes their last-gasp success in Italy shows how much they have improved under Hadden.
White said: "We would have been really disappointed had we gone there and lost.
"The game could have gone either way when it was 10-10 late on.
"Italy are much improved, as they showed by holding Wales, but that victory showed how far we have come.
"They made it very hard for us but in past years we would have lost that game."