It never stopped, despite political pressure to boycott trade with Britain and her colonies areas where it was a major source of income continued to rely on it throughout the War. Smuggling being concentrated around areas with strong connections to Canada and the West Indies. The same was true of the 1812 war. The efforts on a national level to stem the smuggling trade were largely ineffectual as it was tolerated and even encouraged at local level. Trade relations between the US and the UK were normalised by the middle of the 19th c. partially because of improved relations and partially because enforcing boycotts just did not work.
Trading continued after the war ended although very restricted by Great Britain. In 1794-95 the Jay Treaty between the two was signed; it declared America's neutrality in the war between Great Britain and France and opened up trading for America with Britain and it colonies. Britain gained the most from it and the treaty was unpopular in America (particular opposition came from Thomas Jefferson and James Madison), however Washington saw the necessity of avoiding war with Britain and gaining some if negligible ground in trading. Documents relating to the treaty
Maybe someone who knows more about the Jay Treaty can add to this.
This is from the US state department on the Jay Treaty.
Tensions between the United States and Britain remained high after the Revolutionary War as a result of three key issues. British exports flooded U.S. markets, while American exports were blocked by British trade restrictions and tariffs...
The French Revolution led to war between Britain and France in 1793. Divisions emerged in the United States between those who supported the French, including Secretary of State Thomas Jefferson, and those who supported the British, including Secretary of the Treasury Alexander Hamilton. Fearing the repercussions of a war with Britain, President George Washington sided with Hamilton and sent pro-British Chief Justice John Jay to negotiate with the British Government. Jay looked to Hamilton for specific instructions for the treaty. Hamilton recommended an approach that would both stabilize relations with Great Britain and guarantee increased trade between the United States and Great Britain...
The resulting treaty addressed few U.S. interests, and ultimately granted Britain additional rights. The only concessions Jay obtained was a surrender of the northwestern posts (already agreed to in 1783) and a commercial treaty with Great Britain that granted the United States “most favored nation” status, but seriously restricted U.S. commercial access to the British West Indies. All other outstanding issues, including the Canadian-Maine boundary, compensation for pre-revolutionary debts, and British seizures of American ships, were to be resolved by arbitration. Jay even conceded that the British could seize U.S. goods bound for France if they paid for them and could confiscate without payment French goods on American ships.
Jay’s Treaty was immensely unpopular with the American public, but it squeaked through the Senate on a 20 to 10 vote on June 24, 1795. President Washington implemented the treaty in the face of popular disapproval, realizing that it was the price of peace with Great Britain and that it gave the United States valuable time to consolidate and rearm in the event of future conflict.
Edit: added some more sources and also rewording.
Starting in 1651, with the Navigation Acts, England (England in 1651, not Britain until 1707) restricted all foreign nations from directly trading with its colonies without first taking those items through England to be taxed. This was meant to protect and to ensure that England would have a market for its manufactured goods. A major source of contention between the Tories (loyalists who supported the British crown) and Patriots (those who supported American independence) was the precarious economic position that the Americans would be put in if they indeed achieved independence from Great Britain. No longer a British colony, they would be excluded from direct trade to many of the areas that they had primarily traded with before the war for independence. This included the British West Indies, long a source of rum, sugar and molasses. Since the late seventeenth century, and during the first half of the eighteenth, New England merchants had developed a highly important and lucrative trade system with the West Indies. Upon achieving independence from Great Britain, however, these merchants who had relied so heavily upon trade with the West Indies were now completely shut out due to the fact that they were considered a foreign nation now by the British government. Normal trade relations wouldn't be established again until Jay's Treaty of 1795 which once again allowed United States merchants to trade with the British settlements in the West Indies. This agreement would largely remain (albeit with some French protest) until 1812 when war would again break out with Great Britain.
Source: Daniel Vickers and Vince Walsh. Young Men and the Sea: Yankee Seafarers in the Age of Sail. New Haven: Yale University Press, 2005.
EDIT: formatting error