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 forced to use the man's back door.
I had not come through that back door once since leaving for the university, ten years before. I had been teaching on the place going on six years, and I had not been in Pichot’s yard, let alone gone up the back stairs or through that back door.
In A Lesson Before Dying, Grant Wiggins is an educated school teacher who has returned to his small town in Bayonne County, Louisiana, to teach African Americans at the plantation school. However, Wiggins is beset by his own sense of failure to help his people overcome racism and segregation. Initially, he doesn't want to help Jefferson die with dignity because of his own sense of hopelessness and lack of faith. However, it is through his relationship with Jefferson, as the inmate awaits his execution, that both men find a spiritual transformation.
Jefferson is wrongly accused of killing a white liquor store owner, and his public defender calls him a "hog." Jefferson comes to believe that he probably deserves his fate, even though he is innocent. It becomes Grant's task to help Jefferson believe in his own humanity. He tells Jefferson, "I need a hero to tell me what to do, and what kind of man to be. I need you, to teach me that" (49). Through teaching and learning, both characters find redemption, faith, and hope.
Wiggins is not sure he can improve Jefferson's life in the short time before Jefferson is executed. However, Jefferson's godmother, Emma Glenn is a close friend of Grant's aunt. Grant lives with his aunt and persuades Wiggins to visit Jefferson. Ironically, he finds Jefferson just about as reluctant to talk to him as he was to talk to Jefferson.