Greene/Massignani may not be on a terrible high level, but their book at least incorporates Italian sources.
Their version of just what happened after the fall of Tobruk is the standard one outside Rommel's vast fan circle, which included Liddel Hart.
According to Rommel's North Africa Campaign
p 167 events transpired thusly:
Axis plans about Malta needed to be decided now that Tobruk had been secured. Kesselring first arrived in Africa to speak with Rommel on the question of the air units now to be delivered to the Malta operation. "The British are on the run, we should give them no chance to regroup. A later attack on the Nile Delta will need stronger forces and mean higher casualties." Kesselring argued against this, he felt that the air forces needed to rest and Malta needed to fall so that supply lines could be made more secure. But Rommel refused on the grounds that the enemy was in complete crisis. He was informed by the "Gute Quelle" intelligence that the situation of the Middle East Commonwealth forces was so bad that an immediate exploitation of the success would allow the Axis, as Rommel signalled Rintelen, to "destroy the enemy and thus open the way to the hearth of Egypt"
Rommel and Rintelen rarely agreed on anything, so the implication is that Rintelen was all for going ahead with Hercules/C3, which at any rate would have been a mostly Italian operation, with possible dire consequences for Rommel's glorious place in the history books.
Raeder had earlier supported the Malta plan, and if we take Kesselring's views as representative of the Luftwaffe's stand on the matter and if we furthermore assume that whatever Rommel was for Rintelen, representing the OKH, was against, Rommel appears to have been alone in thinking that pressing ahead into Egypt was the right thing to do.
Unsurprisingly the Comando Supremo also favoured Hercules/C3 over attacking into Egypt: the day Tobruk fell Mussolini wrote Hitler and asked for German fuel to be allocated for the upcoming Malta operation.
But as before and since, the German chain of command was utterly chaotic below Hitler. Rommel simply by-passed the opposition and approached Hitler directly:
...Rommel asked the Führer directly for permission and Hitler agreed against the view of Raeder and Kesselring(...)Hitler then wrote the famous letter to Mussolini who was enthusiastic over Rommel's victory and ignored Cavallero's advice. Cavallero was sensitive to the political need not to disagree with Mussolini, and came around. The result was that Rommel was allowed to continue the pursuit of the 8th Army(...)Rommel won the argument, partly due to Hitler and Mussolini's support
In the orthodox version Mussolini only made his flip-flop after Hitler had persuaded him. Greene/Massignani's version of events puts him in a less flattering light.