The New York-based hedge fund said it now owns a 4.39 percent stake in Hess, up from a previous stake of 4 percent.
Over the past several months, Elliott has accused Hess' board of "poor oversight" and said that management was responsible for more than a "decade of failures." It's also pushing to seat five outsiders on the company's board.
Hess rejected Elliott's nominees, putting up its own slate and accusing the firm of trying to disrupt progress it has already made in reshaping itself. It said that Elliott hasn't taken into account how much company shares have risen since it began to shed previous business models.
Earlier this month Hess announced plans to sell its retail gas stations business, along with its energy trading and marketing businesses, as it shifts its focus further toward exploration and production. The company also has announced plans to sell U.S. oil storage terminals and plans to close a New Jersey refinery as it exits the volatile refining business.
Elliott has said that while Hess' moves incorporate parts of its suggestions, they "fall dramatically short of what's needed."
In a presentation posted online Wednesday, Elliott said Hess could be worth as much as $128 per share if the changes it wants are made and the company's management and board are held accountable.
Like those of many other energy companies, Hess shares fell sharply after the recession, but the stock began to rebound last summer and on Wednesday hit a new 52-week high. Since the beginning of this year, the shares have risen about 33 percent.
A Hess spokesman didn't immediately return an email seeking comment.
In afternoon trading Hess shares rose $1.29 to $71.57. Earlier in the session they peaked at $71.73, topping a previous 52-week high of $70.85.