There are few methods of tree stump removal and I will have to choose which of them will work best on my garden. It is a pity that I can’t do something with it and make it into a garden feature and save myself some work.
Lets start with a chainsaw to cut it as low as possible but taking care not to destroy the chainsaw blade with digging into the dirt with it. The stump us too wide to consider tying something around it and hooking to a truck or a car. I can’t get a mechanical digger into the space either as it is too close to the salt water swimming pool. If you can get a JCB or equivalent to dig around it and then push at it from one side. If the stump was a bit smaller the the way to go about it a little simpler. Start about half a metre from the tree stump and start to dig down so there is a circle like a moat around it.This will let you get at some of the roots as you go down. You can hack away at the roots with a sharpened spade, an axe or I have seen a heavy pole with sharpened edge being used. Which will work best depends on the thickness and the density of the roots.
When you have gone down enough you can start to go under the tree stump and eventually do some levering of the stump from under. What would be the best would be a proper stump grinder, no hire places near here so I would have to get a professional in to do the job and that would cost more money. Then you still might have to clean up afterwards yourself with a Pressure washer.
Cheaper options to get at the lower part of the stump.
What can be done is to use a couple of methods or combinations of methods together to do that part of the removal. There is the get it wet and cover with mulch to have it rot away. That will take time though. There is the drill holes down into it method and fill the holes with chemicals to make it deteriorate. Still takes time and depends on the aggressiveness of the chemicals used. The chemicals usually involve weedkiller compounds that are high in nitrates. You have to be very careful with these as they could damage you, your garden or possibly your pets. It is not going to be nice if you get a chemical burn or breathe in some toxic smoke or gas.
Another option could be to light a fire over the stump. Some say that it is good to do that with charcoal like it was a barbecue, others prefer to just get scrap wood. It will take some time also and probably with more than one or two times of lighting a fire.
I have seen a cutting disk that can be fitted to an angle grinder that is used by wood carvers. This is for the wood carving that are larger, not the small delicate stuff on which you would use a dremel tool. It would work in the same way as the big stump grinders and care would have to be taken with having a dangerous tool in the hand.