I think an issue is, and I know I'm parroting people who said this and just stealing if from them, what do any of these nations gain from joining the Axis? Other than a threat to themselves, what would they gain? Argentina and Spain and Chile and so on and so on seem like they'd just be cannon fodder. World powers joining the Axis is what thing, but minor nations joining them is another.
If it were a world without a World War 2, that'd be something else as well, since it'd just be like a Comintern type alliance of Fascists and those who think good relations with Fascists and Militarist states would help them or be in there interest. In a Dieselpunk world, such would be very, very interesting. But it's been said this is for the World War, so I don't see what would benefit these powers to join other than getting attacked.
I think, however, you can strengthen the Axis, and do so just by changing up who is there and what is done.
For example, I think the USSR being in the Axis could be beneficial (should it not lead to Fascist allies withdrawing in disgust from alliance with Germany) since Russia is a world power. If you wish for that to hold, maybe have circumstances make Hitler think breaking off with Russia would not be to his benefit and reconsider invading, or kill Hitler and have him replaced by a pragmatist with a similar thought.
I also think replacing leadership in Italy, and maybe even Japan in Germany, could benefit the Axis. Mussolini did not seem to do well, and seemed to have been more interested in blaming troops than leading a war time state well.
You could also have the war delayed to allow better readiness and strength; Mussolini felt Italy was not prepared, and asked Hitler to hold off starting a war for a few years.
You could also keep some powers at bay. Britain could be kept neutral, or perhaps even somehow made cordial. Russia could as well. And America could be kept out of the war just by having Japan avoid Pearl Harbor, or by having Germany and Italy avoid declaring war, and thus keep the US in the Pacific.