I've done this before (at least 3 times - travel through Changi all the time for work)
You can go through immigration and then back to the transit hotel - the first time I did this was because I wanted my bags w/me (changing clothes after a long flight, etc.). The 2nd time I did it was because when I got to Singapore I realized I didn't have a hotel and the city was absolutely booked (air show)
Go through immigration, get your bag, DO NOT go through customs (which is nothing more than walking out to the receiving hall - all your bags have already been scanned). Instead, after getting your bag immediately turn around and walk back up the ramp (w/your bags), talk to one of the immigration folks (let them know you're going to the transit hotel) and then go up a level to the hotel.
The following morning transfer to the terminal you need to be in, go through immigration again, go through customs, go upstairs, check in.
It's a hassle carrying a lot of bags through the airport (they only have small carts in the terminals) but other than that it works...
Completely separately - if you are landing at a time while the Silk Air counter is still open (I believe their counters are only staffed early morning to early evening) then you'll be able to check in early (up to 24 hours). If you're coming from the US this will be unlikely, unless you are on one of midnight-1AM flights out of SFO or LAX (but then you're likely on Singapore Airlines and then you could check it through).