Cabinet this week approved an additional $14 million in funding, taking the total budget for the plan to $36m.
The first major task, breaching the 30m seal inside the drift, is likely to begin in February.
Three members of the Family Reference Group, which represents some of the families, were at Parliament this morning.
Sonya Rockhouse lost her son Daniel. Bernie Monk's son Michael died, and Anna Osborne's husband Milton also died.
The trio, holding photos of their loved ones, were visibly emotional as Little announced his decision.
Osborne said 29 men went to work that day and no one expected they would never see them again.
"It is a truly amazing day for the families today."
Monk said he always knew that re-entry could be achieved.
"Even when we were in our darkest days, the reason why we stood our ground is, I knew this job could be done," he said.
The families are still hopeful former mine boss Peter Whittall could still face manslaughter charges over the deaths.
Pike River Coal knew that the mine's atmosphere was in the explosive range for a number of days, Monk said.
"They continuously let our men go into that mine and eventually it did blow up, " he said.
Rockhouse said she was unable to say publicly what she wanted to say to Whittall.
"I'm actually speechless when it comes to what to say to him. If I saw him face to face, I'd have one question for him. Why?
"Why did he just walk away and just let 29 men remain down the mine."
In the aftermath of the disaster, WorkSafe put together a case that would have brought 12 charges against Whittall, who was the chief executive of Pike River Coal.
The case was dropped after Whittall agreed to pay $3.41m to the families but the Supreme Court later ruled that was unlawful, and effectively a payment to avoid prosecution.
Little said prosecutions were a matter for police and other agencies such as WorkSafe.
"Police closed their investigation, but not the file, in 2013, so investigations into potential criminal wrongdoing are still open for the police," he said.
Nigel Hampton QC, who represents some of the families, including Rockhouse and Osborne, said it was always his view that recovery of the drift was more significant in some ways than the recovery of bodies, for the purposes of gathering evidence.
"You might well find evidence would possibly indicate the actual cause of the initial explosion, and if that was able to be ascertained then that might help fill in the chain of proof required to possibly mount a manslaughter prosecution against one or more of the management supervision of the mine at the time," he said.
Police Commissioner Mike Bush, who was at today's announcement, did not rule out manslaughter charges.
"Our case is open and everything will be based on evidence. The purpose of us being involved here is, if it's safe, to ensure that we take advantage of any opportunities to examine forensic evidence."
The Herald attempted to speak to Whittall today but staff at the rest home where he works in Wollongong, Australia, said he was unavailable.
Whittall, who left New Zealand in 2014 and is married to Pike River Coal's former financial controller Angela Horne, told Fairfax in an interview in August this year that he felt no guilt.
"Do I feel guilt? No. It is human nature to blame someone," he said.
