The Axis (Germany, Italy, and Japan) were united in that all three wanted to overturn the existing international system and replace it with an entirely new world order. In particular, they (along with the Soviet Union) hated the liberal democracies, whose form or government and economy was seen as decadent and unfair.
To expand the war, you need to expand the Axis which means more countries who are expansionist and desire to end the existing international order.
This is impossible with the world as it was by 1939. We need a POD much earlier than that, but not so early that it butterflies away the end of WWI and other events that caused WWII to happen IOTL.
1) If you want a battle in North America, you basically need Mexico to join the Axis. To get that kind of Mexico, you need a very different Mexico than the one that existed in IOTL. So you need someone different to win the Mexican Civil War. Perhaps the war was so different that Victoriano Huerta became dictator of Mexico. He adopted many of the anti-clericalist measures the PRI later did and had his own cristero war, which he also won. The ending result was an (Italian style) Fascist government that was an early supporter of Mussolini. Later, Mexico received much investment and trade deals with Italy and Germany.
2) As Nazi Germany expanded and prepared for war, Mexico was considered a natural ally along with Italy. Much mutual cooperation between the three happens. Mexico provides oil to Germany in exchange for much technical help and machine tools to expand Mexican industry. Hitler is not too impressed with the half-Indian Mexicans, but he appreciates their geopolitical benefit.
3) As Integralism gains support in Brazil, fascist Mexico helps make Fascism part of normal Latin American culture. Because of that, either Antonio Vargas does not turn against the Integralists and becomes full blown fascist, or domestic changes allows Plinio Salgado to become President in 1938 and turns the country into a dictatorship.
4) Mexico and Brazil join the Anti-Comintern Pact at various times. Perhaps Mexico does in 1937 when Italy does it, and Brazil in 1939. This links them with Germany and Japan.
5) Now comes the hard part - why do Mexico and Brazil declare war on the Allies? Perhaps they do so in June 1940 when it looks like Britain and France will make peace quick (they make the same mistake as Italy). They won't have the same territorial aggrandizement as Italy, but perhaps they think this will give them a better say in world politics with a victorious Germany. A possible collective counter balance to a weakened US might be a good reason. They soon regret their actions.
6) Neither country is able to contribute to the Axis war effort. Both remain too poor and too far away. Royal Navy has other priorities than enforcing an effective blockade against them. However, Germany finds them useful as countries where its U-Boats can find safe harbor. In addition, some Luftwaffe units also transfer (via French airbases in the Western Africa) to defend the ports and support operations.
7) The presence of Axis Brazil and Mexico puts pressure on other Latin countries to be more friendly to the Axis. None join at this point, but the US faces a lot of obstacles to keep other Latin American countries inline.
8) Japan attacks Pearl Harbor in 1941 and wins a resounding victory. Germany declares war a few days later.
9) The US discusses options with its ally Britain. Neither Mexico nor Brazil has declared war on the US at this point. Yet the Mexican dictatorship is extremely unpopular. Many Mexican and Brazilian dissidents live in the US. The US and UK decide that Axis allied nations cannot be tolerated in the Western hemisphere. Dissidents in both countries say a popular democratic government could be installed in both countries.
10) In Spring 1942 the Allies declare war on Mexico and Brazil after diplomatic negotiations fail to make them withdraw from the war and take anti-German/Japanese stances.
11) General Patton is given command of the invasion of Mexico. His forces meet little resistance although a few pitched battles take place. By Summer 1942, US troops enter Mexico City and the Huertista government surrenders. The US establishes a democratic civilian government and eliminates many of the anti-clerical laws. While some US troops remain to keep the peace, they are withdrawn as soon as possible for service elsewhere. The US works out some problems with its doctrine with the learning experience of the Mexico campaign.
12) The invasion of Brazil has the purpose of eliminating the U-Boat threat from the area and preventing it from threatening the US build up to invade Africa and Europe. Negotiations with other South American countries will get them to join the war on the Allied side in exchange for resolution of some disputes in their favor. In Fall 1942, General Eisenhower leads the invasion of Brazil as a combined US/UK naval task force eliminates the Brazilian navy. Forces from Uruguay and Argentina quickly join them. The US publicly offers lenient peace terms promising the restoration of Brazilian democracy. After a few pitched battles, Brazil agrees to a negotiated surrender.
13) The US lands in French Northwest Africa in Winter 1943, incorporating many lessons learned in the previous Mexican/Brazilian campaigns.
I doubt such a timeline could develop in reality, but its at least plausible if you don't look at it too closely.