As for powder stashes, I have not usually had a problem finding something if it has snowed the the past 24 hours. Barring that, your best bet for finding lightly ridden terrain is to hike up to (or past) Headwall. Most people don't bother venturing up there. Be warned, though... if you are an out of shape sea-level dweller then the hike will take you 20+ minutes. Most locals hike up it in like 15 min (or less if they are kids!)
Being a gym/fitness rat, I'm in good shape. However, I'm not used to the elevation. I'll try to make this hike after acclimating a bit.
The morning shuttles run a little more often than that, but they can be a bit of a pain when you are sitting or standing on one for 30+ minutes with all of your gear, making a dozen stops to pick people up. If you are taking the shuttle to the mountain, then catch the first one that you can in the morning.
We are renting a car as well. Sounds like it might just be easier to drive up in the morning rather than stand on the shuttle for 30 mins at a time. I get enough of that bullshit riding the train everyday to work.
Only been to Targhee once. It does get nicer snow, but the day I went was windy as heck. Traveling there and back will consume your entire day (especially if you have to take the shuttle back into town). Nothing wrong with the Targhee except the winding route you have to take to get to it.
I feel like since I'm there I need to hit Taghee up at least once.
Town is interesting enough. If you do a half day during the week then you can check out the gear swap (by the K-mart), do a super-cheap day of riding at the Snow King, or check out the very very yummy bakeries around town.
Thanks for the tips man