I have a dump trailer so the hauling is not a problem.
The hauling effort mention is not hauling the dirt away in a dump trailer, that is the easy part. The hard part is getting it from where it starts into the dump trailer, which is where the wheelbarrow and 5-gallon buckets come in.
When it comes to moving dirt (in Silicon Valley, by the way) I have had great success with day laborers which I get from the parking lot of the local Home Depot. These guys can move somewhere between 1 and 3 yards of dirt per man-day from your excavation. You supply the tools: shovels, picks, buckets and wheelbarrows. Don't stint on these. A $30 pointed shovel will move a lot more dirt over the course of the day than that $15 one on sale. You need one per man, and I would start with 2 or 3 men. Picks are similar, not so many choices in quality grades, but you should start with one per man.
You need one or two wheelbarrows. Get the ones with pneumatic tires, the flat-proof tires are harder to push. Have an extra wheel and tire on hand to take care of the inevitable flats.
The current going rate is $100 per 8 hour day, and I always provide lunch for the men, and have plenty of bottled water. A can or two of Coca-Cola per man day is appreciated.
Don't worry about the men not giving good effort. At the end of the day ask for their phone numbers. They all have cell phones. Call an hour or so later and ask the hard workers to work again the next day. If you have to not re-hire a slacker, just ask one of the good workers to bring a hard-working friend.
If access is good, three guys can load my 6' x 12' x 2' dump trailer to its full dirt capacity (which is only one foot deep, based on an 8000 lb payload for the trailer) in about 1/2 day.
You will need someone on hand to drive the dump trailer away and dump it at least twice a day. For me, this was a 3 hour round trip, so don't forget the driver.
I have done exactly what you are talking about -- excavated under a house, held it up with housejacks and beams, and poured a concrete foundation under it.
You can do it yourself. You do not need a temporary footer outside the current house. In fact, it will be in your way when you come to form up the foundation.
PM me for more details.