This year and last the Cog railway has stayed open in winter. The train stops before treeline, so it's the best of both worlds: plowed road but not too much noise & smoke.
Use the original trailhead - park all the way up at the end of the road (by the second station), cross the tracks and look for a footbridge to the right of the water tower.
Foot traffic is pretty light, so bring snowshoes and give yourself ample time to find & break trail. Axe not needed unless you get lost and wander into a ravine.
Parking in the Hiker Lot at the Base Station is now free (at least for the 2016 season). Access to the trail is different from the description above. Walk to the head of the platform where passengers board the train and cross the tracks. There is a small metal sign that reads "Jewell Trail" (I've posted a photo) and a staircase leading down to the river. There is no bridge across the river (or any sign of there ever being one). If the river is too full to rock hop across (as it was when we were there), your best bet is to return to the boarding area and use the railway utility bridge (marked authorized personnel only, so be sure no railway workers are around). Hike along the tracks until you find an easy place to cross. Then bushwhack back to the trailhead.