As has been mentioned, the recommended long term storage is inflated. Mine are under the spare bed in the guest room.

One thing you asked about, carrying it under your pack using the straps on the pack, might be something to reconsider. If it were a closed cell foam pad, that's probably fine, but, for an inflatable, I wouldn't want that on the bottom. Imagine setting down your inflatable pad on something like a rock, with perhaps a sharp point, and then put your 40 pound pack on top of that? Could be a bad scenario. The Thermarests are great, and I have a Thermarest 4 that's perfect for hiking, but, if you puncture them, good luck making a field repair. That's why on "rough" trips, the old standby is a roll of closed cell foam, or a folded waffle style pad.