After leaving the Falklands on 28 January, Byron was back in Patagonia on 5 February, resupplied his vessels, and took the Florida with him into the Straits of Magellan. She was still with him when he met a French ship (Bougainville’s, on his second Falklands voyage), and was sent home from Port Famine with news of the encounter, details of Byron's progress to date and a regretful remark that ‘Our Ships are too much disabled for the California Voyage'.
The immediate result of the news the Florida brought back was a small party established a British settlement on the Falklands, setting in place a three-way tussle over the islands with France and Spain.
Byron was then, according to his instructions, to make his way into the Pacific, track west, keeping as far south as possible to locate the Great South Land thought to lie somewhere between New Zealand and Patagonia. He was then to proceed to New Albion (Drake's name for California) where he would search for Juan de Fuca’s Strait, thought to be the Pacific outlet of the long-sought Northwest Passage across the top of North America. If the search was successful, the Passage would provide him with a way back to England. Alternatively, if the search was unsuccessful, he could return home across the Pacific via the East Indies and the Cape of Good Hope.
Even a cursory attempt to follow those directions was bound to be time-consuming, and there is already a hint of disinclination on Byron's part to take his time on his investigations. That is, in one way understandable. Scurvy tended to appear around the hundred day mark into a voyage, and the passage through the Straits of Magellan took two months. Heading west in search of the South Land would, almost inevitably, bring on the disease, so it probably made sense to make for a known destination where the crews could recover.
Moreover, there was another consideration. Like everyone before him, the westerly winds Byron encountered once he made his way into the Pacific were always likely to push him too far to the north to seek his next instructed objective.