I understand your excitment and desire to head out to one of our most beautiful parks, however, it will be winter and you should exercise caution.
Online Trail Conditions from Olympic National Park. Use this to assess if the trail, or the road to the trail, you are intereseted in running, is open and safe for travel. Trails are not maintained in the winter, downed trees, landslides and other hazards may close or limit access. Check this out before heading out to the penninsula. Also, lodging is limited, and ferry routes are on winter schedule. I strongly advise going on line and reseraching your idea. I love a good adventure, and have attempted a "plan free" trip to this area in the spring, we ended up doing way more driving than planned.
_____FROM THE WEBSITE________
http://www.nps.gov/olym/planyourvisit/wilderness-t...
~ Road Information: Fall/Winter storms can cause road closures! Always call (360) 565-3131 for the latest Park road conditions.
~The Deer Park Road is CLOSED at the Park boundary for the season.
~The Obstruction Point Road is CLOSED for the season.
~The Staircase Road is CLOSED to vehicles for the season, 1.25 miles from the trailhead.