@GregBelleville I wouldn't worry about using a garden hose. It is just introducing water with minerals into the cooling system. You are going to have to flush out the cooling system with distilled water so putting hose water into it is just adding more steps to the process.
To speed up the process, many people take out the thermostat. Without a thermostat, the cooling system will start circulating immediately. It is still a good idea to let it run for about 10ish minutes to flush everything around. With a thermostat in, you have to wait for the engine to get up to operating temperature before it starts circulating, and then wait 10ish minutes to have everything circulating.
If you want to speed up the flushing process, there is also an engine block drain plug that you can pull out. It can be a pain to get to though so many people skip it and do a couple of extra flushes instead. It is on the driver side of the block. I've done it a few times and I've found that a 6" extension and a breaker bar are the best things to use to get to it. If you don't remove the engine block drain plug, about a liter of fluid is left in the engine block. The point of doing multiple flushes is to get the remaining fluid in the cooling system diluted to the point that it is practically all flushed out. It is nearly impossible to flush it out completely though so the goal is to get it good enough.