With reports suggesting owners New England Sports Ventures are now actively seeking a replacement, the 63-year-old could be living on borrowed time after a woeful six months in charge.
When Kevin Davies headed in Matt Taylor's 43rd-minute free-kick things looked even worse for Hodgson, but a revitalised Fernando Torres volleyed home an equaliser before Cole tapped in his first league goal for the club he joined in the summer.
The result came as a welcome relief to Hodgson after back-to-back Premier League defeats, but it may only give the Reds boss a temporary stay of execution.
After their worst start since the 1953/54 season - when Liverpool were relegated - Hodgson appeared to have turned things around only for a defeat at Newcastle and the dreadful performance at home to Wolves three days ago to raise more questions about his future.
Those became more valid when suggestions began to circulate that NESV were stepping up their search for what will be the club's most important appointment in decades.
The feeling was the American owners were prepared to see out the season with Hodgson, who was brought in three months before their £300million takeover in October, but recent performances and major public relations gaffes have changed their stance.
Hodgson's attempt to defend his criticism of fans who targeted their anger at him in midweek failed to cut any ice with supporters and, more importantly, seem to have had an adverse impact on owner John Henry and chairman Tom Werner.
It did not provide the ideal backdrop for Hodgson's most important match in his short reign and the situation was given extra spice because the man in the opposition dug-out, Owen Coyle, who has transformed Bolton in just 12 months, has been touted as a potential successor.
Add to that the fact captain Steven Gerrard was only fit enough for a place on the bench after cramping up having played 90 minutes against Wolves in his first appearance after six weeks out, and the outlook appeared bleak for Hodgson even before kick-off.
One solitary banner in the Kop read 'Fans' New Year Wish - New Manager', but if Hodgson was expecting his side to show an early reaction to their miserable midweek defeat it did not materialise.
Hodgson was forced to send on Gerrard for Raul Meireles midway through the first half after the Portugal midfielder failed to run off an injury sustained minutes earlier.
Just prior to that Taylor had hacked away Maxi Rodriguez's cross-shot from Torres' looped pass to the far post.
Taylor proved to be even more effective at the other end as, after curling one low free-kick into the side-netting, it was his set-piece which allowed Kevin Davies to head home two minutes before the interval after Fabio Aurelio, making his first league start of the season, had fouled Rodrigo Moreno.
Liverpool could have been ahead by then as Rodriguez headed Dirk Kuyt's deep cross onto the top of the crossbar, while in injury time Lucas Leiva skewed wide from close range.
Torres may have given the impression of a man who did not know where his next goal was coming from, but in one flash of genius it all came flooding back.
The second half was only four minutes old when Gerrard's first-time pass from David Ngog's knockdown picked out Torres on the penalty spot and he clinically volleyed home his first goal since November 10.
Torres instantly looked a transformed player and his next effort from the left of the penalty area whistled past Jussi Jaaskelainen and wide of the far post, while Rodriguez and Gerrard both had close-range shots charged down.
Gerrard breathed new life into his previously prolific partnership with Torres and another chipped pass into the penalty area saw the Spain international fire just wide.
An error by Glen Johnson gifted the ball to substitute Ivan Klasnic late on but the England international was indebted to Reina for holding the fierce shot.
Liverpool finished the game as they should have started it, pressing for victory, and they were rewarded with an injury-time winner.
Gerrard's deep cross was touched back by Rodriguez and substitute Cole tapped in from a couple of yards for his first league goal for Liverpool.
It may have eased the pressure on Hodgson, but even a vastly-improved second half performance may not be enough to convince the owners the club is heading in the right direction.