The strong-running centre has impressed in his three games so far since joining from Salford Red Devils on a three-year deal.
He scored his first try for the East Yorkshire club in Thursday’s 48-8 triumph at former club Huddersfield Giants when he also regularly tormented the home defence with his powerful carries.
Uncapped Griffin, 26, said: “It was good to get that. But it’s still a long process for me.
“I’m still learning to play with all these guys; it’s a new team and tough but I’m happy to get off the mark.
“There’s a few things I need to improve. Defensively I was really strong last week against Catalans but this time it was probably my weak spot so that’s something to work on.”
The former Wakefield Trinity and Leeds Carnegie player felt for depleted Huddersfield who were missing nine regulars.
“Obviously I feel for those boys in there as they do have a lot of injuries but it’s one of those instances where you do just have to turn up,” added Griffin, who hopes to gain international honours soon.
“These are the kind of games where if you underestimate them or let your guard down you can take one on the chin and walk away with a loss.
“We did a professional job and got the points so that’s all that matters.”
But he expects Hull to improve again when St Helens arrive at the KCOM Stadium on Friday.
“The Catalans loss was littered with errors and we were still working on attack,” said Griffin.
“We’ve done a lot of work on that this week and it showed at Huddersfield but we still haven’t put the full package together yet.
“We’ll look to do that but know Saints are a good team.”
Hull coach Lee Radford said: “Josh is a big, physical thing and there’s loads more to come from him, I’m sure. We’ve to get that out of him throughout the year.”