There are approximately three routes to get to Ostrander Hut. Take one of those going there, and then take one of the other two going back out to Badger. When part of the way back, you can spend your extra night camped.
The best route going to the hut is the Bridalveil Creek Trail. That's because it is slightly longer, but more of a steady uphill in the last few miles. The next best is close to there, the Horizon Ridge Trail. It is slightly shorter, but it has more elevation gain and loss, and it has a better view from the top (also some avalanche danger if you get too far east). The Merced Crest Trail is longer with lots of up and down, so it is normally done in one day by stronger ski parties.
Some skiers spend an unexpected night out in the snow. That happens at the bottom of Cardiac Hill, which is 1.5 miles from the hut at the junction of the first two trails. For an expected night out, some people camp at Bridalveil Campground. It is closed for the winter, but it is kind of a protected place for a tent. Plus, you might be able to get liquid water from Bridalveil Creek.
I've also camped a group on the north side of the Glacier Point Road, maybe only 200 yards north, near Bridalveil Creek.
There is also most of a log cabin left in there, if you know where to look.
In general, you want to camp someplace where you have some good wind protection. That is typically in a clearing near the forest, but not right next to a tree. If snow hangs on the tree branches and then dumps, it can damage a tent and maybe hurt somebody. So, I try to make a point of camping at least 20 feet from the nearest tree.