Cole shunned interest from Arsenal to move to Anfield but was sent off on his debut, against the Gunners, and has suffered from fitness problems as he has struggled to make a noticeable impact at the club.
The midfielder will make his first appearance in five weeks when taking the field against Steaua Bucharest in the Europa League on Thursday night and Hodgson believes the support of his team-mates has helped Cole contend with a tricky few months.
"He isn't a naturally confident character. He suffers when things are not going his way," Hodgson said. "Everyone at the club has worked hard to make sure he doesn't take it too hard and take too much on himself, because he's quite happy to take responsibility.
"Sometimes those players can take too much on themselves and push themselves even further down. Everyone at the club, but in particular the senior players and his England colleagues here, has done a great job.
"I don't know how frustrated he has been. But he is a long time with us, a long contract and we know what he's capable of doing. The sooner he produces it of course the better for me, but there is no pressure on him in that respect."