All right; I didn't dig into this too much, but here is my best guess:
I think you could have Sicily fall into Carthagian hands at the end of the OTL Third Sicilian War. In OTL, Carthage had overrun the entire entire island at this point, until Agathocles launched a surprise invasion of the Punic homeland. Although he was defeated, he was able to escape back to Syracuse and negotiate a peace. There are two ways to go about this: either have Agathocles' initial invasion never happen (maybe he gets lost as sea, or his plan is discovered and he is routed) or prevent his escape back to Syracuse. Either way, Carthage comes into full control of Sicily.
I suspect that Sicily is, initially, going to be ruled by a series of proxy Kings and governors initially. However, I could see one of the Greek kings launch an invasion of the island to 'liberate' its inhabitants (Pyrrhus, perhaps?). This might actually be good for Romano-Carthage relations, since neither state is going to want to see the Greeks move further into the Western Mediterranean. Also, up until this era in OTL, Carthage and Rome actually enjoyed pretty good relations with one another. Once the Greeks have been pushed back, I suspect Carthage is going to take a more active hand in administrating Sicily and might annex it directly.
Now, Corsica was already in Carthagian hands, save the interior, and it wouldn't be too hard to gradually increase their control. The same goes for Sardinia which had several Carthagian colonial cities in its south and west. I suspect, with more time (and with Sicily momentarily peaceful) Carthage might settle the northern part of the island as well and bring it fully under their control. After all, their control of Sardinia was never really contested by Rome until the First Punic War. I could definitely see an agreement between Carthage and Rome where Rome acknowledges Carthage's supremacy over Sicily in exchange for an agreement by Carthage to not meddle in the affairs of southern Italy.
The most difficult part of this challenge is Spain. Hamilcar was the general during the Third Sicilian War and I could see the government of Carthage growing concerned over his popularity at the time. Perhaps it would be easy enough to have him dispatched to Spain to secure its resources in an effort to get him out of the city. Now, if we want to assume that the Third Sicilian War ended in 307 BC, that means that Carthage is only going to have roughly 40 years between the beginning of the OTL First Punic War. That might be enough time to make some major inroads in Spain, but I doubt it will be enough time to expand their control over the entire peninsula. Most likely Carthage remains strongest in the South (as it was in OTL) and moves further into the interior, and then forms agreements with the local tribes and states to submit to Carthagian supremacy while they maintain much local autonomy.
Either way, Carthage is now significantly stronger than it was in OTL at the start of the Punic War. Likely, there is going to be growing tensions between Rome and Carthage (Rome was never all that keen on seeing rivals become more powerful than them, no matter the formal good relations between the states). However, should a war break out, Carthage is going to be in a significantly better position. Having full control over Sicily, Sardinia and Corsica is going to give it even more naval supremacy than it held in OTL, and I find it less likely that Rome would even be able to land troops in Sicily. It would also have the manpower of Spain to rely on and might even be able to threaten Italy by land, as well as by sea. Now, we know about the vaunted Roman man power, and how its alliance system made it particularly difficult to bring down, but I think Carthage would have a much better chance in this hypothetical matchup (especially when you consider just how close the conflict was in OTL, anyway).