I think you'd need to build up the past eruptive history of a volcano like Yellowstone first, and look for the ash deposits in the stratigraphy - assuming that it's one of those volcanoes with a history of releasing ash (which I think it is). With enough volume and enough eruptive pressure, the ash could be effective in contributing to shutting out a lot of incoming solar radiation. Gauge their maximum extent from the volcano, and maybe their volumes (taking into account any evidence of deposits that could have been eroded away since deposition). I think that would be one step. Then consider the monitoring data - seismic and gaseous, as well as the structural geology of the volcano - these could give pointers as to the stability of the volcano, and the composition of volcanic gases.
If I remember correctly from my studies of the theories of certain past mass extinctions, it's the ash released into the atmosphere that contributes in a big way to global cooling. Although one of my lecturers had a theory that heat released from the eruption of the Siberian Traps around the time of the Permian-Triassic extinction event resulted in global warming, that subsequently set a chain of events in motion (basically a positive feedback effect that heated up the climate; the theory goes that it involved frozen methane gas hydrates in the ocean melting due to the increase in atmospheric, and oceanic, temperatures, followed by degassing of this methane = more global warming) that resulted in 57% of all families and 83% of all genera becoming extinct, i.e. a major loss of biodiversity.
