The England midfielder has signed a four-year contract after passing his medical, having agreed personal terms on Monday following his release from Chelsea.
He is the second arrival at Anfield after the free-transfer signing of Milan Jovanovic from Standard Liege, with 18-year-old Rangers defender Danny Wilson on the verge of completing a move which could cost up to £5million.
Cole's signing has given a huge lift to spirits at the club but Hodgson knows there is plenty of work still to do after Liverpool's worst finish to a Premier League season - seventh - for over a decade and the departure of manager Rafael Benitez.
"It's not going to be an overnight thing," said the new Reds boss at the club's training camp in Switzerland.
"Last season was a very disappointing season for the club in every respect, culminating in a popular manager leaving.
"You don't change doom and gloom or disenchantment with a signing or two.
"I would never want to dupe the Liverpool public by telling them all is rosy now because Joe Cole has signed.
"There's a lot more work to do, a lot more players are needed and we as a football team have got to make certain that when we take to the field the supporters will see there is something different and that we are making the biggest effort we can make.
"Then we can only hope they back us like they have always backed the club in the past.
"We have to get a good atmosphere back into the club, the Liverpool atmosphere that I've known and seen and experienced - often to my cost - over the years.
"We have to get away from the feeling that everything's not right and the club is going downhill because that isn't the case at all."
Despite the 62-year-old bringing some much-needed stability to the football side of the club there is still much uncertainty off the field with the club having been put up for sale before the end of the season by co-owners Tom Hicks and George Gillett.
Hodgson knows all he can do to help that process is make the club as competitive as possible on the pitch.
"Liverpool has always been famous for the passion and knowledge of its supporters and they know I can't put right the disenchantment with a couple of words or by signing a football player," he added.
"What we can do is hopefully persuade them we are on the right track, that we're trying to get things right again and that we're looking for new investment.
"We know the situation with the owners; the club is for sale and we are trying desperately for the right people to come in and buy the club.
"In the meantime we are trying our level best to do what we can out here on the training field and on the field of play to get results.
"I can only hope that the people who watch us are aware of the situation, that they have their feet on the ground like we've got our feet on the ground and they give us credit if we deserve any for the efforts we make on the field."