Another option are the pay showers in the Ambassador Lounge in Terminal 3. A shower only is 16 SGD (about $13 USD) and it's decent enough.

Print out your return BP in HKG because the Kris Gold and SATS contract lounge are both departure only. They will want to see your hard-copy departure BP (mobile BPs won't work) and Star Gold card.

The SATS lounge opens at 3am and is slightly better than the Kris Gold. There are free massage chairs in the back that are pretty nice. Other food options in T3 is the food court area but not everything is open between midnight at 5am.

SIN is an awesome airport for a MR - almost like doing a domestic turn.