At the base of the Portal Road (in Lone Pine), there is the Whitney Portal Hostel, owned by Doug who also owns the store at the portal. The hostel is about 2 years old. Beds in a dorm room are about USD25 each and the whole room is about USD75. The rooms have from 2-4 bunk beds (4-8 people) and their own full bathrooms (tubs too). It's a nice place and it is fun to hang out with Doug and his wife. Other than the bunk beds, it is more like a simple motel than your typical hostel. I have spent many nights here. If you decide to camp, there are showers at the hostel for USD5 -- good showers with fluffy towels, soap, decent pressure for as long as you want. There is open wifi. You can stop in and just hang or say hi even if you aren't staying there.
Next door is the Old Dow hotel that starts at USD57 for a room with shared bath (down the hall) and about USD70 for a room with a bath. I love this place! It has serious character. If there are just two of you, it is probably a better more comfortable deal than sharing a hostel room with others. I think they will hold a room without a credit card up until 4PM of your intended arrival date (this could have changed)-- no risk option to keep bases covered. You are always welcome to hang at the hostel! I have spent many nights here too.
Next door to the hotel (and owned by the same outfit) is the Dow Motel which starts at about USD100. I have not stayed in the motel structure. It is newer and more typical of well...a motel -- bigger and more modern rooms than the hotel.
Alabama Hills, the area of cool rock formations just adjacent to the town of Lone Pine, is BLM. BLM territory starts just a couple blocks up the Portal Road from the hostel. You could clean up and get the scoop at the hostel, use the laundromat across the street, have a meal in town and bed down BLM style -- it's all within a couple blocks.
Maybe just hitch down (no problem), check out Lone Pine accommodations and sleep BLM if need be. The best coffee is at the Espresso Parlor, a few meters north of the supermarket on the same side of the road. Have breakfast at the Alabama Hills Cafe next to the laundromat. Dunno where you're parked but odds are someone will offer you a ride back up the 395 in the morning -- lots of Europeans around this time of year with big rental cars.
If you want to wait 'til morning to go down to Lone Pine, there is a walk-in campground at the portal. Stays are limited to 24 hours and there are no advance reservations so depending on your arrival time, you should have a reasonable chance of getting a spot. If I remember correctly it is USD20.