I don't know about the actual legal case but the additional anger at the death of another President in an assassination could well prompt a harsher reaction and pressure for him to at least be found guilty. Execution is definitely a possibility but since the attempt was in DC rather than a state I'm not sure what rules would apply.
Note it says in his wiki article "Public outcry over the verdict led to the
Insanity Defense Reform Act of 1984, which altered the rules for consideration of mental illness of defendants in Federal Criminal Court proceedings in the U.S. " so there presumably was a tightening of the rules for the insane defence plead.
One interesting butterfly was that his obsession with Jodie Foster occurred in response to the film Taxi Driver in 1976 and he actually tried going for Jimmy Carter 1st. "Eventually, he settled on a scheme to impress her by assassinating the president, thinking that by achieving a place in history, he would appeal to her as an equal. Hinckley trailed President
Jimmy Carter from state to state, and was arrested in
Nashville, Tennessee, on a firearms charge. " If he had actually tried killing Carter, whether successful or not that could have had some impact on future events.