The Cornish Seal Sanctuary in Gweek usually rescues between 50 and 60 pups over a whole winter but it has already rescued 52.
Staff also expect the influx to continue to the end of March, with Storm Henry preparing to sweep in.
The stormy conditions are preventing staff from releasing seals back into the wild.
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Sanctuary curator Tamara Cooper said many pups were underweight and had been injured by being dashed on rocks.
"They have come in thick and fast over the stormy period and we are expecting more over the coming months," she said.
"Some are only a few months old and they come ashore to rest.
"They are very skinny and need feeding up. We have six ready to go so we are just waiting for a clear spell of weather."
Storm Henry is predicted to bring treacherous conditions and high winds of up to 90mph to the north of Britain today.
Storm Gertrude brought gusts of more than 100mph in places as well as heavy rain and snow.
Wildlife experts said an "extraordinary" number of dead dolphins have also washed up on Cornwall's shores this month due to recent weather systems.
Abby Crosby, Marine Conservation Officer at Cornwall Wildlife Trust, said: "We've had 19 different cetaceans, that's dolphins porpoises or whales, recorded when the last ten yearly average is 11. So we're looking at almost double the January average."
"It's a really sad situation, however it's an invaluable opportunity for us to examine these carcasses further to try and work out how they died and find out more about the biology and ecology of these amazing creatures."

Dijkhuizen's assistant, Roy Hendriksen, has also left the Championship club.
Brentford won only two of their eight league games under the Dutchman, 43, and are 19th in the table.
Former Blackburn, Coventry and Everton midfielder Carsley, 41, had been working as manager of Brentford's development squad.
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Dijkhuizen was appointed head coach by Brentford in June, following the controversial departure of previous boss Mark Warburton.
It was confirmed in February that Warburton, who went on to lead the Bees into last season's play-offs, would leave the club at the end of the campaign.
Warburton had guided Brentford to promotion from League One the previous season.
A former striker, Dijkhuizen had previously been manager at Excelsior in the Netherlands, where his side finished 15th in the Dutch Eredivisie.
Ex-Republic of Ireland midfielder Carsley will be assisted by Paul Williams and will take charge of the Bees for the first time when they host one of his former clubs, Birmingham City, on Tuesday.
"The club believes Lee deserves the opportunity to step up and help shape Brentford FC's future on the pitch," said a statement from Brentford.
Dijkhuizen's departure comes just days after he said he "felt no pressure" about his job despite Brentford's stuttering start to 2015-16.
His final game in charge at Griffin Park was a 2-1 defeat by Sheffield Wednesday, the club's third at home in all competitions this season.
"A chaotic start to English management comes to an abrupt end for Marinus Dijkhuizen.
"During his brief tenure, he has had to deal with matters out of his control, such as problems with the pitch leading to a postponed fixture.
"On the pitch it has been far from a successful spell, with last season's Championship play-off semi-finalists winning just two of their opening eight league games.
"In his defence, Dijkhuizen has been unlucky with injuries and has had to deal with the departure of some key players. His undoing may well be that he was indeed too honest in interviews with the media, something that may well have irked owner Matthew Benham.
"Question marks also hovered over whether he had a strong enough relationship and rapport with players who he didn't sign."
A seal sanctuary in Cornwall has seen a huge rise in the number of injured pups in the wake of Storm Gertrude.
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Brentford have sacked head coach Marinus Dijkhuizen after nine matches and placed Lee Carsley in charge until the end of the season.