Normally the transfer desk will help with this (don't go to the gate).

Which flight from ICN-SIN are you on?

I can see the last SIN-HKT flight is at 6:30pm and by the sounds of it you're on SQ611 that arrives at 6:25pm?

You'd need to be at the SilkAir transfer desk in Terminal 2 at least 40 minutes prior to departure (and not be traveling with checked luggage to even be considered) - so that would be 5:50pm as check-in would close for the 6:30pm HKT flight then.

Given the Earliest SQ611 has landed in the last three months is 5:49pm I don't like your chances but if you do land around that time it wouldn't hurt to ask.

The only other thing to point out is the later Jetstar flight at 7:50pm... obviously SQ/MI won't book you onto it however nothings stopping you booking it yourself with prices ranging for $65 to $300 for the 2 hour flight. Assuming you only have carry on and don't need a visa for Thailand then you can mobile checkin and go directly to the Jetstar gate. This would only work however if the the SIN-HKT flight is the last in the booking as skipping the following mornings MI flight will obviously result in all other flights on that booking being canceled.