Former Melbourne Storm loose forward Hinchliffe, in his third season with the West Yorkshire club, is out of contract at the end of the season and unsure about his future.
However, now fellow Australian Woolford has arrived in the country, the 33-year-old hopes to soon be able to discuss plans.
He admits he would like to carry on with the Fartowners who have won their last four games to climb into the top-eight for the first time this season.
“He only arrived on Friday so I’ll let him get over his jetlag first!” Hinchcliffe said, when asked about contract talks.
“But I’ll hopefully speak to him in the next week or so to see what the future is and what he’s going to do with the squad and whether I’m a part of it.
“It’s been a bit of a tough three years over here at stages but I still feel that I’ve got a bit of unfinished business.
“What’s happened over the last month I do honestly feel that the people we have got involved means there can be some good things to come from us and I want to be a part of that.”
Hinchcliffe has been performing well for Huddersfield ahead of Thursday’s Ladbrokes Challenge Cup quarter-final against Catalans Dragons.
It marks Woolford’s first official game in charge after flying into the UK - he arrived just hours before last Friday’s win at Salford - but Hinchcliffe already knows the former Canberra Raiders star.
“I played against him a few times back home,” he recalled about the 43-year-old.
“He was a grubby little hooker! A tough little bugger. A real creative number nine. “He was actually captain at the Raiders whilst I was there playing reserve grade for a couple of years.
“He was a real competitor and a winner the way he went about things. He’ll be a good addition to the club.
“I started as a hooker myself, made my debut there and played 20 odd first-grade games there for Canberra but he’d moved on to the Dragons by then.
“I learned a fair bit off him. I trained with him a fair bit and he helped when I was trying to learn my craft as a hooker.
“It’s funny now that we end up back together. We’ve bought into what he’s said and we just need to keep continuing and improving on those things.”