Nobody advocating for free market healthcare is expecting people to shop around in emergencies. People are expected to shop around for good insurance plans which will cover emergency situations to their satisfaction. That is why many free market healthcare supporters also support removing the incentives for employer-provided insurance plans:
- remove the tax exemption for employer-provided insurance
- add a tax exemption for individual/family insurance, including a rebate for the amount of money payroll (social security and medicare) taxes would affect the amount of money available for that insurance
This way, people don't lose their insurance when they lose their job, and people have more insurance options. This would also likely have a beneficial affect on insurance competition.