First you need to look at the motives of Tokugawa Ieyasu. As the leader who finally conquered Japan and become the Shogun, he was well aware of the destabilizing nature of Western technology. After all, he himself used massed firearms to defeat Tokugawa Ieyasu in the Battle of Nagashino, sweeping away one of the most skilled and feared traditional Samurai armies of the day in the process.
Closing Japan was to limit destabilizing influences and allow the Shogunate the time and space to consolidate their regime and create a stable environment for the Japanese economy and society to recover from essentially 200 years of civil war. Keeping people inside Japan was part of this, since sailors, merchants and adventurers exposed to foreign influences could bring back ideas, which was just as dangerous as Western traders bringing firearms, cannon and other goods which Japanese society either did not have or would have difficulty making (one of the reasons the firearms ban was so effective in the Tokugawa Shogunate revolved around the very few skilled gunsmiths in Japan and the essentially craft nature of Japanese manufacture at the time. In Europe at this time, the Serenìsima Republic Vèneta had the Arsenal, building cannons and warships on what was essentially an assembly line).
So for a contrafactual that allows Tokugawa Ieyasu to encourage foreign adventure, perhaps he wins the Battle of Nagashino, but it is not a decisive victory. At that point in history, there were only three contenders left for the Shogunate, Tokugawa Ieyasu, Oda Nobunaga and Takeda Katsuyoui. Realizing that no side had enough power to gain a decisive victory and the nation could be embroiled in continuing war, Tokugawa Ieyasu calls a meeting of the three clans. The offer is simple: to prevent an endless and essentially unwinnable war, if the Oda and Takeda clans would agree to resettle in the new lands across the sea, the Tokugawa clan would agree to lay down arms against them and finance the building of fleets for the journey. For those members of the clan who could not go due to age or other reasons, the Tokugawa clan would effectively adopt them, providing for an honourable life of peace in Japan.
The two other clans would be rather dubious of this, but the attraction of striking out to new lands, being free of the restrictions of land and resources which crimped Japanese life and the end of the constant cycle of war would be attractive enough for a large fraction of the Oda and Takeda clans leadership to take the offer seriously.
Tokugawa Ieyasu allows scout ships to cross and come back to report, which makes the plan all the more exciting for two clans. They finally agree to send thousands of people across the oceans to resettle. Tokugawa Ieyasu sets up shipyards and funding to do so, with the only provision that the people who depart are never to return to Japan. To keep their new lands and prevent bloodshed, the leaders of the others two clans agree, and for the next century, mass fleets of Japanese vessels set across the Pacific and invade the west coast of North America. While there is a general agreement to prevent back flow to Japan, there is enough leakage from adventures (both sailing from Japan and North American Settlers sailing back) to keep word of the colonies and the various opportunities and dangers being faced by the colonists alive in people's heads.
Relived of the threat of civil war and with the population both busily employed and some population pressures being relived, the Tokugawa Shogunate begins the transition to a peaceful and orderly Japanese society. After the first wave of settlers have departed and a period of time elapsed, future Shoguns might be inclined to start sending expeditions to North America to examine the colonies and begin trade with the Home Islands in the 1700's.
