Can this man appeal for something he has already been charged with by the jury?

Most appeals are not based on the original defense argument. They are usually based on technical problems with the trial, poor legal representation, evidentiary issues, flawed jury instructions, misinterpretation of the law, etc. No jury's decision is final and conclusive until all avenues of appeal have been exhausted. That's why a death penalty sentence is more expensive to the state than a sentence of life in prison.