To connect on a Jetstar Asia flight from SQ, you will need to clear immigration, collect your checked bags, proceed to Terminal 1 and check in at 3K's counter 2 hours before your 3K flight to BKK.
Please note that there is no 3K Transfer desk within the Transit area.
For showers, you could consider the following options.
1) Proceed to Terminal 2 Transit hotel or Plaza Premium Lounge after disembarking from SQ. Both are located above the garden just before F50 gate. After the showers, clear Immigration, collect your bags at SATS Lost and Found counter and proceed for check in for 3K in Terminal 1.
2) Clear Immigration and collect your bags and proceed to check in at 3K counters. Re-enter the Immigration area. Proceed for showers at either the Rainforest Lounge (near Gate C1) or Transit Hotel Terminal 1 (above Gate D41), depending on location of the boarding gate for 3K.
The showers will cost approx S$8.50 and is inclusive of shower amenities/towels and after shower beverage.
Cheers
J777