If Hitler died in the fall of 1941 Goering would probably become the nominal head of State. The Reichstag would vote the Reichmarshall Chancellor of Germany. The office of president had been merged by Hitler with that of chancellor on the death of Von Hindenburg in Aug 1934. Since Hitler never bothered to write a new constitution the Weimar one was still in force. Since Goering wouldn't be Fuehrer since that title was so associated with Hitler a figure head president might have been appointed or the office just ignored. Hitler's cabinet would remain in place along with the military hierarchy.
Actually, while the Nazis never bothered to write a new constitution or carry out the promised Reichsreform, Hitler stipulated in his decrees on the succession that Göring would succeed him in all offices in state and party. The Law on the Succession of the Führer and Reich Chancellor (Gesetz über die Nachfolge des Führers und Reichskanzlers) of 13 December 1934 gave Hitler the right to appoint his own successor. On 19 December 1934 he did that when he signed a decree that made Göring his heir in all his offices and further stated that all members of the Party, the government, the SA, the SS and the military would have to swear an oath of fealty to him. One exemplar of the decree remained with Hitler, a second with Lammers and a third with Blomberg. Göring was only informed orally. The other Reich ministers were only officially informed about the arrangement in 1936, though Göring unsurprisingly did not miss the chance to informally spread about it because it was something to brag about, and thus the inner circle was already aware by then. On 23 April 1938 Hitler signed another decree that stated the same (and also stipulated that Hess would manage the Party under Göring if something happened to Hitler, which was essentially what Hess was at the time already doing for Hitler).
He made the succession
arrangement public on 1 September 1939:
' Should anything happen to me in the struggle then my first successor is Party Comrade Goring; should anything happen to Party Comrade Goring my next successor is Party Comrade Hess. You would then be under obligation to give to them as Fuhrer the same blind loyalty and obedience as to myself. Should anything happen to Party Comrade Hess, then by law the Senate will be called, and will choose from its midst the most worthy - that is to say the bravest - successor.'
Hitler signed another decree in 1941 shortly before Barbarossa, which restated that Göring would be his deputy if he was incapacitated and serve as his successor. The only difference was that Hess was removed from the succession arrangement due to his flight to Scotland. He only split the offices in his political testament in 1945, since there was nothing left to inherit anyway and Göring had, as he saw it, betrayed him. But both his Tabletalk and draft laws from the Reich Chancellery from 1941 and the Reich Interior Ministry from 1939 on the creation of a Nazi Senate post-war show he didn't intend for the Führer title to be restricted to him alone. And Göring's ego and desire for power is definitely big enough not to settle for anything less.
I apologize if it wasn't clear, this isn't directed at you in specific, I was referring to a general misconception in these alternate history circles, which fits right along others like the Apolitical Speer.
On the eve of Barbarossa, Göring asked Heydrich to provide a small manual so that the troops would know who to 'put up against the wall'. And during a meeting with Count Ciano, he bragged that Nazi starvation policies would kill 30 million Slavs. The Hunger Plan - or more precisely Hungerpolitik - was in many ways devised by people who at the time were part of his bailiwick. He thought very highly of Herbert Backe. And Göring was an important factor in the Nazis' anti-Jewish policy until about 1942. As Phil Blood shows in one of his books, the Luftwaffe also had death squads of its own.
Never mind the fact that, from the Nazi perspective, there was no contradiction between exterminating the Jews and working those considered 'useful enough' to death. It's literally in the Wannsee Protocol. And Jews employed in the armaments industry in the General Government were initially spared from Aktion Reinhardt in OTL. It's pertinent to note that by autumn 1941 the Heer, SS-police, and Waffen-SS/Kommandostab Reichsführer-SS etc. are already murdering Soviet Jews en masse. What had not been decided yet was the complete extermination of all European Jews
during the war, though as the Nazis' own documents show, the Madagascar Plan had already been intended to be genocidal in nature. By the time the Wannsee Conference was convened the decisions had been made. It was an implementation conference, and Heydrich based his commission on Göring's authority.