Hey Chelsea! Great question.
I do a little bit of trail running, but typically with less vertical gain and mileage than the Six-Pack of Peaks.
As long as you don’t encounter more than patches of snow, trail runners will work fine. That’s typically what I use on day hikes even. If there are some stretches of snow, a pair of micro-spikes can be very useful — but as long as there are not winter mountaineering conditions. In those conditions, boots, crampons, ice axe, helmet, and the knowledge of how to use them all properly are essential.
The good news is that we’re approaching the end of that period pretty rapidly, so by May or June the trails will be established even though there may be patches of snow remaining.
I would suggest that the gear you bring on your trail runs will be sufficient, but if you’re hiking, it will take longer, so you’ll need more water and food.
And although many of these trails are quite popular, you’re still traveling in the wilderness, so carry the 10 Essentials.