I would say that yes, Hannibal's chances where greatly improved by crossing the Alps. For this answer I am going to focus on a few key factors about Hannibal's Italian campaign and its political impact within the Roman Senate. Otherwise this answer would be an extremely long description of the Second Punic War.

First Hannibal's crossing of the Alps was not a spur of the moment decision. Both Roman and Carthaginian generals identified the Iberian Peninsula as a major theater in the war. The Iberian peninsula and modern day southern Spain supplied the Carthaginians with silver, manpower, food and shipyards. The Romans knew Iberia was the way into Northern Africa and they needed to stop its industrial output from fueling the Carthaginian navy.

Here is a map to help us all visualize what I’m talking about

Second, a large part of Hannibal's strategy was based around the fact that he correctly identified a major weakness within the Roman armies disposition and he had identified a key weakness in the Northern Gaulic tribes that bordered the Roman Italian lands. A common misconception is that through out Roman history, the entire length of Italy was Roman. However Rome had unified Italy in a series of wars and engagements, creating a group of Roman allies called the Socii. It wasn't until the social war of 91 BCE that Roman Italian allies would be granted citizenship within Rome. Similarly the Northern part of modern Italy and Southern France, Cisalpine Gaul was inhabited by Gaulic tribes Rome had previously attempted to expand its influence over. As a result several Gaulic tribes in Northern Italy where in open revolt. Hannibal knew these Gaulic and Italian allies had bones to pick with Rome and that they could just as easily become enemies. Hannibal also knew that if pressed into war against Carthage, Rome would remove most of its forces from Northern Italy to face them. As they did not perceive the Gauls as great of a threat as they where.

So far we know Hannibal's motivation was to attack a perceived weak spot within the Roman disposition. Additionally it was very important to the Carthaginians to keep the war away from their agricultural heartland in Africa. Hannibal's strategy was certainly seen as a way to achieve that goal and defeat the Roman Republic. In 219 Hannibal put this plan into action by sieging the city of Saguntum. Saguntum was a hellenized ally of Rome, the border between Roman and Carthaginian influence within Iberia was said to be at the river Ebro and Saguntum was right within the disputed area. Hannibal used the execution of Carthaginian supporters within the city to lay a bloody 8 month siege that would end with the sacking and destruction of the city and its inhabitants. Saguntum during those 8 months would repeatedly call to the city of Rome for help. Rome did not answer with any help.

So we have set the stage for one of the most dramatic moments in human history when after a little political waggling the Roman Senate declares war against Carthage in the spring of 218 BCE. Starting one of the bloodiest conflicts in human history, The Second Punic war had officially begun.

Hannibal's first action is to march his army North into the Pyrenees. He splits his army into three columns and continues to advance through the Pyrenees conquering the local tribes. This was a pretty bad slog actually, Hannibal suffered significant losses securing the Pyrenees. However by the end of it Hannibal had expanded Carthaginian control over the Pyrenees. Now we had the beginnings of the route that would take Hannibal into the heart of Roman Italy. The Greek historian Polybius had this to say about the Hannibal's Pyrenee's campaign.

These measures satisfactorily accomplished while he was in winter quarters, and the security of Libya and Iberia being sufficiently provided for; when the appointed day arrived, Hannibal got his army in motion, which consisted of ninety thousand infantry and about twelve thousand cavalry. After crossing the Iber, he set about subduing the tribes of the Ilurgetes and Bargusii, as well as the Aerenosii and Andosini, as far as the Pyrenees. When he had reduced all this country under his power, and taken certain towns by storm, which he did with unexpected rapidity, though not without severe fighting and serious loss; he left Hanno in chief command of all the district north of the Iber, and with absolute authority over the Burgusii, who were the people that gave him most uneasiness on account of their friendly feeling towards Rome. He then detached from his army ten thousand foot and a thousand horse for the service of Hanno,—to whom also he entrusted the heavy baggage of the troops that were to accompany himself,—and the same number to go to their own land. The object of this last measure was twofold: he thereby left a certain number of well-affected persons behind him; and also held out to the others a hope of returning home, both to those Iberians who were to accompany him on his march, and to those also who for the present were to remain at home, so that there might be a general alacrity to join him if he were ever in want of a reinforcement. He then set his remaining troops in motion unencumbered by heavy baggage, fifty thousand infantry and nine thousand cavalry, and led them through the Pyrenees to the passage of the river Rhone. The army was not so much numerous, as highly efficient, and in an extraordinary state of physical training from their continuous battles with the Iberians.

Now I realize that's very long and dry, but the key thing to draw from it is that Hannibal pacified the tribes within the Pyrenees and left some men behind to garrison the territory and because he believed them liabilities during the crossing. Hannibal was aware he was taking a large risk and he clearly wanted to scrap every advantage he could during the crossing. Once Hannibal had garrisoned his new holdings and readied his troops, he marched to the river Rhone. Now the Rhone had a problem with it, the Greek city of Massalia stood at the mouth of the river where it flows into the sea. Massalia was a Roman ally and wealthy marine merchant city, making a coastal crossing of the Rhone a problem for Hannibal. So Hannibal marched north before crossing the river instead of advancing along the coast.

Now the Roman army hadn't been doing nothing this entire time. The Roman's after their declaration of war in the spring of 218 they split their army into two primary pieces. One would reinforce Roman Sicily and the other would arrive at Massalia to advance along the Gaulic coast. While the Romans knew Hannibal had crossed the Ebro river, they where surprised to find him north on the Rhone river in September when they arrived in Massalia. One of the reasons the Romans where so surprised to find Hannibal advancing through Gaulic territory was that they assumed the Gauls would harass the Carthaginians like they did the Romans. However Hannibal had negotiated with the Gauls as he moved through their territory, telling them he came for Rome alone. Hannibal's political prowess allowed him to navigate a difficult environment filled with potential enemies and to turn many of them into allies. So Hannibal had already begun to exploit the Gaulic weakness of the Romans, thus allowing him new avenues of travel and attack. The Romans lead by Gnaeus Scipio and Publius Scipio (Father of the famous Africanus), upon discovering the new Carthaginian position dispatched a scouting party north along the river to find Hannibal. They would in fact discover the Carthaginian scouting party dispatched to do the same. The Numidian cavalry that made up the scouting party was defeated by the Romans and pushed all the way back to their camp. Shortly before this engagement, Hannibal had crossed the Rhone in a battle against the Gaulic ally of Rome, the Volcae. This battle is called the Battle of Rhone River and a brief summary of it is that Volcae occupied the eastern bank of the river to oppose the Punic army. Hannibal split his army under the cover of night and sent one detachment farther north along the river, he then advanced across the river with both parts of his army, once the Volcae had engaged his infantry crossing the river on the east bank, his second force revealed themselves and split up. Both pinning the Gauls against the infantry crossing the river and the rest burning down the Gaulic camp. This broke the Volcae and won Hannibal his river crossing.

Upon learning of the location of the Punic camp on the River Rhone, the Romans forces advanced north from Massalia along the river to meet Hannibal and his army. Now this is a key moment in the war, Hannibal is presented with a clear choice. Abandon his current plan and engage the Roman force meant to invade Iberia in Gaul or disappear north and continue the crossing. Many would pin blame for the loss of the second Punic war here, on Hannibal. As we know now Iberia would be essential to winning the war and the Carthaginians without Hannibal would suffer a series of crushing defeats in Iberia until the Roman pushed into northern Africa. Of course that is mostly something we know with the benefit of hindsight. Hannibal had left garrisoning forces in Iberia, he left trusted family members and generals with a strong navy. Of course what happened is famous. Hannibal, rather then meet the Scipios in battle, once again fled north along a mystery path that dropped him at the foot of the Alps in Autumn. The Roman army unable to find the Punic army to force their big battle they wanted, continued along with their plans to invade Iberia. While Publius left with some forces to warn the Roman Senate of the danger coming to northern Italy.

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