When I built my house the header tank (in this case 5000 gallons) was about 100m (300') away from the house and the fall was about 10m (30'); those are still the current dimensions. From the tank to the house the pipe is 2" diameter poly pipe (as known in Oz, but called ABS pipe in the US I think) and the piping internal to the house is 1/2" copper.
When the system was first commissioned we had exactly the same problem that Bobert is describing; no flow at the (bottom) business end of all this piping. The local plumber came out with a stirrup pump (the sort you put into a bucket but has a leg outside the bucket with a stirrup that you stand on while pumping by hand) and, after we'd selectively shut various inlet ballcocks so that we were dealing with known and short lengths of internal piping, it took less than 5 gallons to pump enough water to remove the airlock in the copper.
I had tried using an electric rotary pump but it hadn't worked; it took a positive displacement pump mechanism to create enough force to move enough volume of water to wholly displace the airlock. Most plumbers around my area (rural, with a 25,000 population city in the middle) have a stirrup pump which they'll lend to "known" customers for precisely this sort of situation.
Bobert's long hose sounds as though it may have had at least one airlock in it at the beginning of his exercise and I don't know (yet) if he's eliminated them; once the hose has nothing but water in it, siphon action will deal with small rises in its length providing of course that the "lower" end really is lower than the "higher end".
On the infrequent occasions I've had to get a serious length of garden hose full of water, without a decent head or a pump, I've found it easiest to just coil the hose and chuck the whole coil into the creek. Bobert, you could use your header tank; airlocks can be manually manipulated to the end of the coil or you could use one of those battery-powered drill pumps that John has mentioned. When they've been removed, put a bung in each end of the hose and lay it out through your snake paddock.