With respect to the FT'ers that say that Singapore can be done in a couple of days (and it's really true), I'd like to play devil's advocate and make some 'off the beaten path' suggestions for Singapore; it sounds like you would like to be in the countryside and surprisingly there are such places in Singapore.

You can try the Sungei Buloh Wetland Reserve. It's up in the northeastern part of Singapore, should be good for a nice long hike, felt like being in the jungle (albeit with footpaths). Not too too far away is an outfit called Bollywood Farms (the area is called Lim Chu Kang) - it's an organic farm run by one of the movers and shakers in Singapore society but is not a dilletante operation. Anyhow it should be good for a tour, also they have a restaurant on site (would you believe it's called Poison Ivy, a pun on the owner's name?) that makes good use of their own veggies. In the north/northeast there is Pulau Ubin - it's a separate island on the Singapore side of the Johor Straits, other side is Malaysia. This island is a little like the place that time and Singapore forgot - locals like going there for hiking, also to rent a bicycle and ride around and about. BTW Outward Bound Singapore uses Ubin as a site for some of their training, if this helps.

If you were to go to these places, you can add two more days to your visit. Take note that mass transit doesn't conveniently go to these places (especially Bollywood Farms) so add some time for getting around. I have to warn you that I'm not a nature person so even though I've been to Pulau Ubin and Sungei Buloh and enjoyed it, it may not suit you. However local street reputation of all three places is pretty good, so it's worth a look.