One obvious way to have raised production was to avoid inefficiency. The problem is that while this included simple incompetence, it also resulted from frequent changes of plan, some sensible but others less so. The challenge is to plan ahead.
In 1939, one problem was that the plans were preparing for a war around 1943-1944. Thus, as described by Tooze, the first step was to raise ammunition production. This accounts for most of the initial increase in war production.
At the same time, to free space for U-boat construction, two H-class battleships and an aircraft carrier were broken up. Several civil liners were also abandoned. Another aircraft carrier was stopped, later restarted and then abandoned. One large heavy cruiser was sold to Russia and a second, Seydlitz, which had been launched pre-war, was never completed.
U-boat production rose steadily until conventional U-boats were defeated in 1943. The need to build different designs after 1943 must have again reduced output.
From July 1940, the luftwaffe and navy received higher priority but the highest priority was preparation for Sealion, which slowed the completion of Bismark and Tirpitz by removing labour. I am not sure when army production gained higher priority but I think it was some way into the short campaign planned to start in June 1941.
Looking closer at aircraft output, there are huge well documented loses because it was planned to replace the Bf-110 by the Me-210 and the He-111 by the Ju-288 and He-177 from 1941. The production of the old aircraft was tapered off but had to be increased after their replacements were found not to be ready for production.
There was also a long running dispute between Milch and the operational commanders on the introduction of improved variants. Naturally both were correct. The best solution was probably what the US did for the B-29 by running production lines without too many interuptions while the airforce modified the aircraft (battle of Kansas). The Luftwaffe also followed that method which may account for some of the later increases in production. Oddly, the Luftwaffe may have had to add midification packs more as they shed men to the new (and generally unsuccessful) luftwaffe field divisions.
Finally, if you want real inefficiency, look at the occupied countries (I am currently reading Mazower's Hitler's Empire)! However, the most obvious example of getting Ukrainian industry working, might have only raised output from 1942.