The Lanarkshire side narrowly missed out on the title after finishing second, two points behind the eventual Scottish Championship winners Dundee.
But they booked their place in the play-off final after Tony Andreu scored the only goal of the game against Falkirk to ensure Hamilton progressed 2-1 on aggregate.
And Neil admitted his side are keen not to pass up a second chance of promotion to the Scottish Premiership.
"I was aggrieved that we didn't win the league and we have used that as motivation in terms of saying that we maybe blew that opportunity but now you have the second bite at the cherry," he said.
"What you don't want to do is let it bypass you twice. You know how hurt you were after the first time and so let's make sure you don't have that feeling again.
"To get a crack at Hibs is great. We have had a great journey this season and I didn't want it to end today."
Sunday's goalscorer Andreu believes all the pressure is on Hibernian, who have won only one of their last 18 league games.
"It is the most important goal I have scored so I am delighted," he said. "We are looking forward to the Hibs game, to show how good we are.
"These are the most important games in the careers of most of the players, me included, so we have to be confident.
"We don't have to fear Hibs, we just have to play our game and we will see what happens.
"The pressure is on Hibs, they haven't had a great season. We have nothing to lose."
Falkirk boss Gary Holt took the positives from his young side's performance, insisting it has given him a reason to be optimistic about the club's future.
He said: "I don't see it as a negative. After going a goal behind we showed true courage for a young squad and the younger members of the team stood up to be counted and gave it their all and that is all I can ask for.
"It is disappointing? No. I have loved every minute of what the boys have given me this season and how they have managed to cope with being at the top end of the table and pushing for the title and then the play-offs.
"It is something to build on and we will keep going."