Grant does not want to go to the jail to talk to Jefferson because Grant does not really see the point and he believes that he has better things to do. Grant is a grown man with a job and a love interest. He simply doesn't want to spend his time going to the jail for something that he sees as a hopeless cause. Grant is being asked to help Jefferson die as a man rather than while thinking of himself as a hog.
"I don't want them to kill no hog," she said. "I want a man to go to that chair, on his own two feet."
Grant doesn't see the point because either way Jefferson is going to die.
Another reason Grant is reluctant to help out is because the situation would require him to ask for help from Henri Pichot, and Grant absolutely doesn't want to go there. Henri reminds Grant of oppression as Grant has always been...
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