Actually they do. This one is very short walk to Terminal 2.
https://www.google.com.sg/maps/place/Crowne+Plaza+Changi+Airport/@1.3586541,103.9880221,15z/data=!4m5!3m4!1s0x0:0x74a0170f1cc50445!8m2!3d1.3586541!4d103.9880221
From T2 to T3 there's a train connecting or you can walk from T3 too.
But I think that's a bit expansive.
There's this other one which is one train station away which should be cheaper.
https://www.google.com.sg/maps/place/Capri+by+Fraser+Changi+City/@1.353377,103.9697402,14z/data=!4m12!1m6!3m5!1s0x0:0x74a0170f1cc50445!2sCrowne+Plaza+Changi+Airport!8m2!3d1.3586541!4d103.9880221!3m4!1s0x0:0x46bd008b22cf731e!8m2!3d1.3341014!4d103.9636606
The other alternative is pay to use a lounge if you have access to one or find a spot and slip into a sleeping bag. I'll be writing about a lounge I went sometime.