Free upgrades are almost non-existent in SQ's repertoire. I wouldn't count on it
Technically you wouldn't get access to SQ's Silver Kris lounge but give it a shot. They may let you in.
On a previous flight, I flew to SFO via HKG on SQ, SIN-HKG in the Suites and HKG-SFO in Y. I'm always very pleasant with the check-in agents
and the agent at SQ's Suites/First Class Check-in Terminal was also very nice! I don't know what it means but she placed a golden sticker with the Singapore Airlines "kris" on my HKG-SFO Y boarding pass. When I was transiting in HKG I asked if I could visit Silverkris there and showed them both my boarding passes. Didn't think I'd get in since my onward trip was in Y but the lounge agents comments to each about the golden sticker and they let me in.
In case they deny you entry, the paid Ambassador Transit Lounges in T2/3 have showers.
You might also want to check out the other amenities at Changi airport, they're all free. Some that come to mind:
- Butterfly Garden in T3
- Orchid Garden in T2
- Video game and entertainment deck in T2 (diagonally above the Orchid Garden)
- An actual in cinema in T2
- Swimming pool in T1
- massage chairs in every terminal