Melaka, Malaysia, Singapore
If you're looking for something to do out of the city, why not go on a day tour to Malacca, across the border in Malaysia? Tour East run day tours for $80SGD, which include bus fares, a buffet lunch and sightseeing in Malacca. Excellent value.
The city of Melaka is one of Malaysia's most historical destinations with Dutch and Chinese buildings dating back to the 17th century and later the British with their colonial architecture.
Melaka can be done on as part of a packaged bus trip tour from Singapore. It's a whole day trip of around 12 hours.
Not sure one can claim Malaysia is off the beaten path. But you can very easily go from Singapore. Be it for the day or longer. With JB being only about 40 minutes away and KL only about 5 hours it gives you a good excuse to see another place on the same trip.
I used my second weekend in Singapore to go to Kuala Lumpur on the overnight bus and back on the overnight train.