Going from London to Auckland I get routing through Singapore for ~£1300. This is good because I'd like to visit Singapore.

However, 6 hours isn't enough in the airport. Really I'd like 12-48 hours.

But when I look at separate flights it's £821 first leg, then £828 for the next.

How can I get that first price? What can I do to search for a route with a long layover in Singapore?

Is there a term to help me search for this? Singapore airlines seem to have the most common routing so my next move will be to call them but I'd like to know the technique for other routes too if there is one.