After winning what war did the us become an empire? what colonies did it gain?

The United States became an empire in 1898 when we annexed Hawai’i and the Spanish overseas empire. When we annexed Alaska, as a non-contiguous territory, we were on the way to empire. When we annexed distant tropical islands we were there. Over the next decades we created Panama and annexed part of it, we occupied Nicaragua, Haiti, the Dominican Republic, and we intervened in Mexico, China, and Russia. After WWII we adopted neo-imperial, indirect methods and engaged in horribly bloody, proxy wars with the Soviets. Since the end of the Cold War, we have returned to more direct interventions. But, because these do not involve direct annexation of foreign territories, they are not truly imperial. They need a prefix, neo-, semi-, post-, or something else.

I hope it helps!