Napoleon fled (if that's the right word) in some style, travelling with a "suite" that included three generals, two French counts and countesses and their four children, ten army officers, a doctor, two cooks and 26 other servants along with the imperial dinner service and silver plate and "several boatloads of luggage". So this clearly wasn't act of a panicked man attempting to out run, or to keep a low profile to evade, his pursuers. At the same time the size of his party couldn't have escaped notice and would have made his eventual capture inevitable when he was actively hunted, once Louis XVII came to power.
It appears that Napoleon's ultimate plan was to "retire" to the United States, with the aim of convincing the allies that it would be better for all concerned for him to be on the other side of the Atlantic. It certainly seems that the interrim French government (prior to the Bourbon restoration) were happy to see him leave because they had applied to the British for a passport that would have allowed him to go. To that end, Napoleon headed to Rochfort where a couple of frigates were waiting.
There was possibly a spy among Napoleon's travelling companions because, on the 30th of June, the British had received an unsigned letter notifying them of the Emperor's exact intentions and warning them to prevent his escape. As a consequence, the 74-gun Bellerophon was keeping station at Rochefort under orders to prevent the French ships there from leaving.
However, even with the British blockade, Napoleon hadn't given up all hope. On the 10th of July, He tried to bluff his way passed the British navy.
A letter dictated by Napoleon (but signed by Grand Marshal Count Bertrand) was sent to Captain Maitland. It stated that Napoleon had abdicated the throne of France with the intention of seeking asylum in the United States of America, sailing on the frigates. It continued on to say that Napoleon was expecting a passport from the British government which, it stated, had "been promised to him". The bearers of this letter hinted that Napoleon still had a power base in central and southern France and it would be in the best interests of the British to allow him to leave rather than risk continuing the conflict. The implication being that the Captain may be acting against the interests of the British government if he prevented Napoleon's departure. Additionally, they also enquired if Napoleon would be allowed to leave upon a neutral vessel (rather than on the French warships).
Unfortunately for the Emperor, Maitland was aware that his government had not issued (nor intended to issue) any passport that would allow Napoleon to travel to America. Since Maitland wasn't sure that he could stop all of the potential vessels that Napoleon might attempt an escape on, he in turn bluffed in order to keep Napoleon in port.
Maitland's reply stated that since Britain and France were still at war he couldn't allow any ship to leave the port, however, he'd passed on the Emperor's letter to his admiral and was waiting for advice. This allowed him to pretend that approval might be on its way. In fact, he'd already received orders, from Admiral Henry Hotham, not only to prevent Napoleon's escape but to get him on board the Bellerophon and return him to Britain.
Another plan was hatched by Napoleon's staff, with the aim of sacrificing one of the frigates to engage and delay the Bellerophon while the other fought off the smaller British ships. However, the Captain of the frigate Saale (which Napoleon's party was on board) refused to be a part of that plan and, with the Royalist flag raised in nearby La Rochelle, Napoleon's options were almost exhausted. After a final check that his passport hadn't arrived, he made a decision to surrender himself on board the Bellerophon.
In retrospect, Napoleon had actually squandered a couple of good chances to escape to America. If he'd set sail immediately upon his arrival (on July 3rd), it's possible that the two frigates would have been able to evade the larger, slower British battleship, which was on its own, and make it to the open ocean. However, he waited for five days while he organised his household and during that time two additional British ships arrived. Then later, on July 13th, he was visited by his brother Joseph, who'd chartered an American ship (anchored in the Gironde), and who offered to take Napoleon to New York. Napoleon declined and told him to make his own escape. Joseph successfully reached the States six weeks later.
The Surrender of Napoleon, Sir F.L. Maitland, revised by W.K. Dickson (Blackwood, 1904)
Billy Ruffian, the Bellerophon and the downfall of Napoleon, D. Cordingly (Bloomsbury, 2003)