Long answer: The effect of volcanic eruptions on climate is somewhat overblown in popular culture. In recorded history, we only have a handful of VEI7 eruptions, and only one within the range of well-recorded history (that one being Tambora). Tambora is known to have been followed by a particularly severe summer, but it was also concurrent with climatic minimums due to the "Little Ice Age" and the solar cycle. Some other VEI7 eruptions are known, but it's highly speculative how severe any sort of climatic was response, both locally and globally.
A supervolcano would be a VEI8 eruption. There are two known VEI8 eruptions in the history of humanity, both well before recorded history. People have claimed that one of these, the Lake Toba eruption, is responsible for a severe bottleneck in human population, but work has shown that a) global ash distribution is identifiable, b) wildlife populations don't seem to have been too affected, and c) even relatively close populations don't seem to have collapsed.
In other words, the largest eruption in the climatic record was incapable of producing an extinction-level event. A Yellowstone caldera eruption wouldn't be all that much larger. For anyone in the ash cloud, it would be problematic for the short term, but that's a relatively short term issue (1-2 years). More serious would be the loss of productive farmland in the US for perhaps a decade or so, but that wouldn't be anywhere near extinction-level for humanity.