The ex-St Helens stand-off - who has forged a promising partnership with Australian scrum-half Aidan Sezer - has extended his deal at John Smith's Stadium until the end of 2021.
As one of the club's biggest out-of-contract players, it is a crucial piece of business for Giants ahead of the Super League return on August 2.
"Gasky is obviously an important part of the squad we’re building moving forward, it’s a real relief he’s committed for another 12 months," said Woolford.
"It’s been really pleasing to see the combination he and Seze have formed in such a short period of time, and I’m excited to see their natural progression and improvement once we get back playing."
Since joining from Bradford Bulls in 2017, Gaskell has scored 22 tries in 80 appearances for the Fartowners and was voted players and supporters' player of the year last term.
"Our aim before we broke up was always the play offs and we started the season really good," he said, with the squad due to resume practice soon.
"We know it’s been along time since we’ve trained or played with each other, but hopefully once we return we can keep that form going.
"It’s obviously been a difficult time for everyone during this current pandemic, so to get a deal sorted before we go back to training was the main focus, so I could fully concentrate on having a successful season when we eventually return."
With Super League shutdown since mid-March and players on furlough, Gaskell, 29, conceded: "It’s been pretty tough to be fair with not having any access to anything.
"I managed to get hold of some equipment to tick over with, and also been going for runs trying to keep my fitness up.
"I think everyone’s excited to be back playing; it’s been along time now sat at home not really knowing what’s going on but I think with the latest news of the return it’s given everyone something to look forward to and getting back into routine."
Giants narrowly avoided relegation last term but had lost just once in Super League before the pandemic hit in 2020.
"I think the biggest difference in the squad this year is the players believe we can compete with any other team in the league, and I think it’s shown the first five rounds," said Gaskell.
"But the biggest part now is coming back and hit the ground running we all know it’s going to be tough but if we have the right attitude there’s no reason why we can’t push for that top four."