This weekend we came really close to buying a Honda Odyssey so we would have plenty of room for camping gear. That would have meant a car payment :-( We regained our sanity and decided that we just have to make our 2010 Toyota Corolla work. We need to fit 2 dog crates (they take up the whole back seat), a large tent, 2 sleeping cots, ice chest, card table, little bike trailer, clothes, stove and cooking supplies, 2 bikes plus more! We have a bike rack and are checking into a roof rack. Any suggestions for packing a small car?
Small risk to this but ditch the spare tire, assuming you have CAA/AAA and aren't anywhere really off the beaten path. That'll add a bit of cargo space. Keep a tire repair kit on hand so you can fix a flat.
I also agree you need to streamline your gear: not cots. Think about ditching the dog crates and get "doggie seat belts."
Forget the card table. Get some of those folding fabric chairs, use your cooler as a table.
Do you have 60/40 split folding rear seats? That might help a bit.
Get a hitch mount cargo "shelf."

Come to think of it, putting the cooler on that rack while you're parked gives you a bit of a higher table.
This may not be in the cards for this trip but look at ditching the Corolla. Not for a clown-sized Honda Odyssey but something like a Toyota Matrix or Hyundai Elantra Touring (or 2004-2006 Elantra GT hatchback). The added room of a hatchback opens up a tonne of possibilities. (
a bunch of other cars are mentioned on the MMM blog)
Oh, and this will be probably weirdest suggestion: if you don't have split folding rear seats but they were optional on your car, visit the auto wrecker and see if you can retrofit your vehicle. I've done this type of thing and it can be economical. Ditto accessories - they might have a Corolla on the lot with a roof rack.
FYI: I car camped with a dog and girlfriend in a Toyota Tercel. (it did have the split rear seats though... helped with larger items)