Just like most people in first-world countries (third-world countries are third-world because they're much farther from the "problem" of technological unemployment) can get by now working a low-wage part-time job if they don't mind having no electricity or running water, in the future you'll be able to get by on a low-wage part-time job if you don't mind living exactly as you are now.
What makes you think those low wage part time jobs will still exist? It's highly likely that they'll increasingly be automated out of existence.
You're right.
Those jobs eventually won't exist, because they'll be automated. Instead, other - easier, higher paying - jobs will exist as people desire greater and greater comfort in their lives and hence start wanting tiny, really easy things done.
Proofreading my comments here for readability takes a few extra seconds/minutes. Shall I hire someone to do it? How about hiring someone to adjust the feng shui of my sentence arrangement? Perhaps, as mentioned above, I'll simply pay someone a satoshi to get out of my way since I'm in a hurry. Maybe someone to follow me around at a distance with binoculars to periodically make sure there's nothing embarrassing stuck in my hair or on my face/clothes.
In a world where having just a bit a skill allows you live like a wealthy person does today because everything is insanely cheap, I might want to pay for all sorts of things. There will still be relatively rich and relatively poor, of course, and it will naturally usually be the richer paying the poorer to do such things, which is how the relatively poor will be able to get the tiny amount of money they need to live in what is today considered great comfort.
Yes, the world you describe is certainly one possible outcome, but I don't believe it's in any way assured.
It's assured by basic economic logic. If nothing needs doing, it's because, well, nothing needs doing. No one wants for anything. If you have a solar-powered house that does everything automatically, including growing food and preparing it and serving it, you'll have no housework (no jobs) at all to do, but is this a bad situation? (Perhaps you'll pay for someone to chew it for you so you can continue your dinnertime chat uninterrupted?) The economy where truly nothing needs doing is like that solar-powered Inspector Gadget house. And transitory phases on the way to that ideal just involve things like 2-minute workweeks to pay for the ultra-cheap goods and (almost all automated) services we need to enjoy our present standard of living.
I fear the result will be business as usual, and a market economy which excludes an increasingly large proportion of society from meaningful participation in society. Followed by a bloody revolution.
The market isn't excluding anyone from participating in society; government barriers are. In a free market, anyone who can provide any service that anyone values at $4 per hour (or $3, or $2, or $1) can get a wage just slightly below that. Some people simply cannot provide services that are valued any higher than $1 per hour, but that is nothing novel or unique to high technology; imagine the lot of such a person 200 or 2000 years ago. They would be reliant on charity, or living in whatever is considered relative squalor at that point in history. At least in a free market society they'd be able to make a tiny amount of money working, in addition to whatever charity (or if you like, welfare) they receive, instead of nothing...and in the future when things are fantastically cheap they may be earning enough to live better than any of us are now.
This is a gigantic boon,
especially for the poor and unskilled. Far from being a cause for concern, it is rather the solution to those very types of concerns.