The word layover has no clearly defined meaning (despite the Bourdain series).

A connection is under 24 hours (sometimes extended to first available flight if later, regardless of whether seats are available on that flight) for international, usually under 4 hours (with the same possible extension) for domestic itineraries.

OP seems to want a STOPOVER. Some (usually not deep discount) fares permit this and some permit it for a fee (perhaps about $100)....or you can just book multicity (essentially three one way fares on the same ticket) or even do an open jaws ticket plus a separate OW ticket for DPS to TPE (or vv).