Six hours makes searching easy if you are crossing continents because you will need a mid-ocean stop and there are not many of those. After that you just know which big hubs are roughly en route and stitch it together. You are reverting to a much earlier era of flight. Is it doable still? Yes, but don't expect it to be easy or quick. In other words: the way I would search this is not to search but rather know which route I want to take because there are not that many ways to hop from one end to the other end of the world with six hours tops. And then buy many tickets. You can check flight times on Google Flights.
In practice this will (almost always) mean that you fly through Honolulu or Rejkjavik. Over the Atlantic, some very eastern Canadian airports are less than six hours from Europe, for example YYT to LHR is only five hours and YHZ to LHR is 5:55 and that's about it. These, however, will be very expensive except the seasonal WestJet flight from YYT to LGW. Everything else is out, for example the YHZ-FRA Condor flight is too long, 6:40.
Flying from the US to Europe via Rejkjavik is very heavily marketed by Icelandair, you will have no problems booking a trip. Toronto and New York just to name two large hubs are less than six hours away and all of Europe is a short flight from KEF.
Now let's turn to the Pacific Ocean, as I mentioned you need to stop in Honolulu, that's evident. San Francisco-Honolulu is the shortest and even that's 5:23-5:40 depending on which airline you ask.
In my experience buying a JFK-(west coast)-HNL plane ticket and a HNL-(Asia) plane ticket might be cheaper than a single ticket.
However, HNL-(Asia) is longer than six hours. You have two choices. Either you give up on the Pacific Ocean and go the other way around: you need to start out to Rejkjavik. Then, again we look for our next tentpole: Central Asia is not particularly rich in hub airports. Your choices are Novosibirsk, Astana, Tashkent and New Delhi.
Let's see how each work out, some European hub, Moscow, Novosibirsk, Khabarovsk (5:15) or Vladivostok (exactly 6:0), Tokyo. It'd be easier if Keflavik-Moscow existed but alas, it doesn't seem to.
Going in the middle is actually the least stops this way 'round: Frankfurt, Tashkent or Astana, Beijing, Tokyo. This route is very remarkable because almost everything in the world except Australia is less than a six hour flight from one of the stops: New York-Rejkjavik-Frankfurt-Astana-Beijing.
New Delhi: something Europe, Istanbul, New Delhi, Guangzhou, Tokyo. Again, it'd be better if Rejkjavik-Istanbul direct existed but it doesn't.)
Back to the Pacific Ocean. As I mentioned you need to San Francisco, Honolulu and then you need to use the United Airlines Island Hopper:
The longest leg is Honolulu to Majuro is 05:04, Guam to Tokyo is three hours or so. Although it's more stops than the Iceland-Central Asia route I described above, it's a much easier trip since it's one flight for five of the stops.
Edit: finally, Jonathan Reez found another route. The background is: in old times Anchorage and Gander used to be the refuelling airports and I dismissed them in chat as not being serviced internationally any more. Turns out, you can fly from Anchorage to Petropavlovsk once a week in the summer with Yakutia and on to Tokyo with the same airline. The ANC-PKC leg leaves on Monday, arrives on Tuesday. The PKC-NRT legs leaves on Saturday... but this is certainly the fewest stops. You can fly to Tokyo with S7 on Thursdays via Vladivostok, though. To get to ANC I'd recommend doing NYC-DEN-ANC on United so you don't have >6 hour legs.