While a few hundred people in one area could allow the species to survive (a minimal viable population), it might take centuries or even many thousands of years for them to approximate the pre-event level of civilization with such a small starting point.
The Americas or Australia had a starting populations of that order of magnitude, but it took them something like 10,000 years before they reached a reasonable level of technological civilization - longer than people who were just reproducing existing discoveries would take, but quantity matters, not just quality, when it comes to reproducing civilization's accomplishments.
You'd really like to have more like 100,000 working age adults to preserve enough of the right mix of specialized modern skills important to a modern civilization, and to have critical masses of people in each area (perhaps several dozen to a few hundred) sufficient to manage until they can meet each other.
This large number is necessary both because some modern technology skills like medical and scientific and engineering and cultural specialties are quite rare, and because lots of knowledge isn't present in any book or other fixed record and has to be passed on from one master to an apprentice to be preserved successfully.
No historical society starts to display any meaningful level of technological sophistication before they have cities with 100,000 or more people.
Also, some fudge factor is necessary above what a planned colonization maintaining modern levels of technology would require. This is because there will be huge language barriers involved in uniting the people that remain, because many people in rural areas will be lost, because many adults will be too traumatized to function even if they are otherwise skilled and able bodied, and because it is likely that the survivors won't be able to manage to immediately unite into a single global society so the critical mass needs to be reached with less than the total remaining global population.
People in very rural areas, while perhaps more self-sufficient, are already so thin on the ground that they barely have enough population density to function and a significant drop in population would undermine that.
At one person in 10,000 surviving, there would be 800,000 survivors, perhaps two-thirds to three-quarters of whom are adults capable of working, and at one person in 100,000 surviving, there would be 80,000 survivors.
In Tokyo, there are 37,274,000 people, so that would be 373 survivors in a one in 100,000 scenario, and 3,727 in a one in 10,000 scenario.
If it takes 20 survivors to last until connecting with other urban centers, that takes 2,000,000 people in a city in a one in 100,000 scenario and there are 252 cities that large - with people in less densely populated cities often not making it and reducing the global critical mass below a high civilization maintaining level in the early first couple of years before society could rebuild. You would probably lose more than half of the 80,000 survivors early on in that scenario, and it would take longer for them to establish coordinated groups of even tens of thousands of work capable adults.
In a one in 10,000 scenario, a city of 2,000,000 would have 200 survivors, and a city as small as 200,000 could still have enough survivors to make contact with other survivors (and there would be many thousands of such cities). You would lose a much smaller share of the initial 800,000 survivors and would get to a critical mass of people coordinated in a single cooperating society much sooner.
For the reasons discussed above, I think that with 1 in 100,000 survivors you are going to take many, many generations to rebuild and you will lose a much larger percentage of the initial survivors early on. But with 1 in 10,000 survivors, a larger share of the initial group of 800,000 will make it, and a group with a critical mass of about 100,000 adults will come together quickly enough to avoid excessive knowledge loss before this critical mass can be assembled.
So, something that was 99.99% lethal could be recovered from in a single generation, while recovery from something 99.999% lethal would take much, much longer to rebuild after the event, possibly centuries or thousands of years.
One in ten thousand is still a quite small survival rate.
For example, it is on the same order of magnitude as the number of people who are born with one of the thirty or so monogenetic disorders known as muscular dystrophy, or the number of people with IQs of 160 or more. It would be twevle or so times smaller than the number of elected officials (federal, state or local) in the United States relative to its population, and similar in number to the number of adults with net worths of $100 million or more in the United States. The tallest one in 10,000 women are about 6'2" or more (one in 1000 are 6' or more), and the tallest one in 10,000 men are about 6'8" or more (one in 1000 are 6'6" or more).