The loop road was mostly plowed and snow free, but still closed to cars. We started from the visitor center and road walked a few miles before heading out on the at times paved trail to Beauty Cave (not that we could see too much pavement under the snow patches). We continued on the faint trail east from there for a good ways before turning south and going pure XC across the lava to the edge of Little Praire, where we found a nice camp.
The next day we wandered around south and east of Little Praire exploring the flows before diagonalling back NW to pick up the so called wilderness trail near Split Butte. We followed this (or rather the snow on top of it) back to the end of the paved road and hiked this back to the car.
So in summary we got away from the pavement a good ways, though there is so much more to explore. A dream trip would be to traverse the whole monument complex south to north.
Ideally one would go a we bit later in the thaw, so that you'd have snow to melt for water but not need snowshoes. We found occasional bits of standing water, but never enough in one spot to make refilling practical.
It's far from everything, but very worth the drive.