Hangul script as a writing system has a number of advantages, it's believed to have a high degree of readability and helped the expansion of literacy in Korea. It was artificially created under the orders of Sejong the Great and despite opposition from the Confucian elite is today predominantly used in South Korea and North Korea (in the latter, Chinese characters have been fazed from use entirely) and used also in regions of China settled by Koreans.
Basically, I am wondering whether the Hangul script could have expanded to non-Korean languages. As it is so closely linked to the Korean language and nation it is difficult to imagine scenarios where it would get very far, except in unique instances. It is arguably a much better aesthetic fit for Asian languages than the Latin script used in many countries today.
Here is a discussion on the use of hangul for the Shanghainese language.
The obvious would be an expansion of the Korean empire, which would be somewhat difficult but not impossible in the Joseon dynasty. The likely target of expansion would be Manchuria, though a naval expansion would also be interesting. A Korean Taiwan or Okinawa, for instance.
Then there is the possibility of a Chinese king adopting the script for similar reasons to King Sejong, though there would be severe national backlash. Perhaps the Manchus would attempt to do so, though they did have their own script. Regardless, the script would have to be modified, but I think that in the right set of circumstances it could spread in China with an effect on raising literacy among the peasantry. Perhaps adopted by a group an anti-central-government group like the Taiping.
An alternative vector may be an alteration in the expansion of the Japanese empire over Korea. The Japanese encouraged the use of Hangul in the late 19th century in order to draw Korea away from the Chinese yoke, but later banned its use. Perhaps instead the Japanese could have sought to create a hierachy of literacies, with Japanese citizens using kanji, katakana and hiragana while subject peoples using hangul. In this way, it could be applied to regions such as Taiwan and Manchukuo.
Perhaps a bit unlikely. Another possibility is the French, who sought trade relations with Korea in the 1860's but were rebuffed. If, instead, Korea had entered into diplomatic and trade relations with France it could have seen the hangul script being somewhat better known. It's ease of use and natural fit for Asian languages might see an adapted form of the script being introduced to French Indochina, later resulting in Vietnamese written in the Hangul script. With luck and timing, it could start a fashion for it among other Asian nations seeing it as an Asian alternative to the Latin alphabet. Then again, many of the countries I'm thinking about in this regard had/have their own traditional scripts.
Any other possibilities?