What are possible solutions when promoting competition is not applicable? Usually, more competition means lower prices and higher quality as less effective businesses go out of business. However, it appears that it's not always the case. When there're a lot of health insurance companies, each of them has relatively little bargaining power with service providers (hospitals) and, as a result, prices are significantly higher compared to a hypothetical government-funded single-payer system. Is there any free-market solution? What libertarian scholars (they exist, don't they?) would suggest?

There's another example, social networks. Facebook was selling user private data without them realizing it but never went bankrupt or lost a lot of users. It's evident to me that it's due to its oligopolistic nature. You can't nationalize Facebook or do something similar to what is proposed for health care by American presidential candidates like Bernie Sanders. You also can't promote the creation of many new tiny Facebooks because the social network user wants to have all their friends, family members, colleagues, favorite celebrities, etc. available in one place so people would unwittingly gradually monopolize "the social network market" back by opting for the network with more users than its competitors.

Are there any studies that found any umbrella solutions (preferably, ones that don't involve socialism) for cases when simply promoting competition doesn't work?
There is one universal "solution" to monopolism without government intervention.

Just allow the monopoly to do its worst. It will give very strong incentives to find ways to satisfy needs of potential clients of the monopoly in ways that would diminish market force of the monopoly. Basically it's bet that human creativity will eventually tame (or even kill) the monopoly. The most radical example of such creativity is a creation of an invention that makes the whole monopoly obsolete. Another example of creativity: It's also possible that different people buy services/goods of the monopoly for different reasons, so there would be strong incentives to develop imperfect substitutes for different categories of potential clients of the monopoly. Of course there are many more other ways to outsmart the monopoly, I just gave some abstract examples of it.

Real life example: US shale oil boom as response to monopoly of OPEC