I believe the trail you are referring to leaves Lone Pine Campground and goes up the south side of Lone Pine Creek to Whitney portal. Since there is currently snow above about 7000 feet, much of that trail would be covered with snow.
Around Lone Pine not that much is available while the roads are closed.
You can walk up the road to Whitney Portal from near the road closed area. It may not feel like a great hike but would be a good workout for 5-6 hours.
You could also hike to the church up Tuttle Creek
Take a look at Rick Kent's photos. It is an interesting place and would take a few hours or so.
http://www.rickkent.net/ViewerPlus/viewer.asp?FolderID=264You could hike up to the Waterfall up Lone Pine Creek. That looks like an interesting place but it is not far. See the pictures @
http://www.mt-whitney.info/viewtopic.php?t=424