Searching for flights with stopovers (the term used to describe a long layover, usually over 24 hours, where you retrieve your bags and check them back in when you depart) is not an easy task.
Airline policies are very variable in this respect. Some airlines do encourage stopovers in their "home city", pushed to do so by local authorities. They will often have special deals for hotels, transfers, etc.
Cathay Pacific used to do that a lot for stopovers in Hong Kong, don't know if they still do. Some of the Middle-East airlines do so as well for a stopover in their home city.
Some airlines on the other hand have restrictions on stopovers (as they are seen as a way for business travellers to pay less for flights), including:
- allowing stopovers only on some fares (the more expensive ones, of course)
- limiting the number of stopovers
- limiting where you can have a stopover
- adding a charge for each stopover
Doing a generic search is not easy, especially if you have flexibility on the number of days you are willing to stay, as this adds yet another dimension to the search.
The find such fares, there are multiple methods:
on search engines which allow "multi-city" searches, enter all the segments. This will usually try to find fares that match the total flight rather than the individual segments.
if you are interested in a specific A-B-C combination but are flexible on the dates, you could use tools sur as Expertflyer to:
- Look up the available fares on A-C
- Check the fare rules for each fare and keep the lowest fares that allow stopovers in B.
- Find the matching fare class for each fare (may be different on the different segments)
- Find which A-B and B-C flights have availability for the given fare class
It's quite time-consuming though.
The way you would search depends a bit on whether you are set on a specific itinerary or are willing to consider multiple options.
Note that even if the airline does not charge more for the stopover directly, there may be additional taxes (1 arrival + 1 departure often costs more than 1 transit in many airports).