Take a short snowy jaunt along the Wonderland Trail from Longmire to Cougar Rock Campground. This stretch of the Wonderland Trail parallels both the Longmire to Paradise Road and the Nisqually River, and has a mild incline all the way from Longmire to the campground.
You can extend your hike by continuing past the campground to the junction with the Nisqually River, where there's a nice view of Mount Rainier on a clear day. More experienced snowshoers with snow navigation and avalanche skills may continue across the river if the bridge to Carter and Madcap Falls is up, but note that there is potential avalanche danger on that route.